A Modular SU (5) Littlest Seesaw

We extend the littlest modular seesaw to a Grand Uniﬁed scenario based on SU (5) endowed with three modular S 4 symmetries. We leverage symmetry protected zeroes in the leptonic and down quark sectors to suppress deviations to the littlest modular seesaw predictions, but not contributions to the quark mixing. The model is supplemented by two weighton ﬁelds, such that the hierarchical nature of the charged-lepton masses, as well as the quark masses and mixing, stem from the content and symmetries of the model, rather than a hierarchical nature of the Yukawa coeﬃcients


Introduction
The Standard Model (SM), though successful, does not provide any insight into the origin of fermion families, nor their curious and hierarchical pattern of masses and mixing parameters, which, including also the contrasting neutrino sector, is known as the flavour problem.One approach to the flavour problem is based on flavour symmetries, where the approximate tri-bimaximal nature of large solar and atmospheric neutrino mixing, together with smaller reactor mixing, motivates the use of simple non-Abelian discrete symmetries such as A 4 , S 4 and A 5 [1][2][3].Modular invariance [4,5] can provide the origin of such symmetries in terms of levels N = 3, 4, 5, respectively, of the quotient group of the modular group with its principle congruence subgroup [6], leading to very predictive models of flavour.Of special relevance are the fixed points or stabilizers [7,8] where for certain values for the modulus, subgroups of the modular transformations are preserved.This has been generalised to the case of multiple modular symmetries in [9][10][11][12].In order to account for the mass hierarchy of the fermions, an extra singlet field called a weighton [13,14] may be introduced, without the requirement of an extra symmetry.The origin of all quark and lepton masses and mixing may be addressed by combining Grand unified theories (GUTs) with modular symmetry groups, for example SU (5) GUT models at level 2 [15,16], level 3 [17][18][19] and level 4 [20][21][22] 1 .
In this paper we consider a SU (5) GUT model with three modular S 4 symmetries, which can lead to a predictive Littlest Modular Seesaw model of leptons [44], while at the same time accommodating the quark masses and CKM mixing parameters.In order to address provide a natural explanation of mass and mixing hierarchies, we employ two weighton fields [13,14], resulting in a triangular form of hierarchical down-type and quark and charged lepton Yukawa matrices as in [21].The resulting hierarchical triangular forms preserve the successes of the Littlest Seesaw model while allowing downtype contributions to the CKM angles, with the weightons providing the hierarchical suppressions in all charged fermion sectors, including the up-type quark Yukawa matrix.We present benchmark points which demonstrate the viability of the approach, and show how higher order operators may be controlled by judicious use of the modular weights across all three S 4 sectors.
The layout of the remainder of the paper is as follows: we start by quickly going through the modular framework in Section 2, followed by a brief introduction of the SU (5) embedding of the model, shown in Section 3. Following, we present the model in Section 4, including the numerical results.Finally, we conclude in section 5.

(Multiple) Modular Invariance Framework
The Littlest Modular Seesaw relies on multiple modular symmetries to impose the CSD(n) structure, with bi-triplet flavons which acquire vacuum expectation values (vevs) in such a way that the three S 4 modular symmetries are broken down to a diagonal S 4 subgroup which effectively mimics a single modular symmetry, with different moduli, depending on the invariant considered [9].The inclusion of flavons (with non-zero vevs) will spontaneously break the modular symmetry, such that it is no longer always possible to perform a modular group action γ such that only the fundamental domain may be considered [46].In the low-energy theory (when the modular symmetry is broken), the whole domain is relevant, and we can make use of all of the different fixed points [7,8].This is also possible to understand in the context of multiple modular symmetries, by recalling that the bi-triplets break the multiple modular symmetries into a diagonal subgroup.As such, the group action will transform all moduli simultaneously, and consequently it is no longer possible to send all moduli to the fundamental domain in general.In this section we briefly review the modular symmetry framework and the extension to multiple modular symmetries.
The modular group Γ is defined by three generators [6,47], which obey S 2 = R, and (ST ) 3 = R 2 = 1, together with T R = RT .A group element γ acts on the modulus τ (with Im(τ ) > 0) via fractional linear transformations: a, b, c, d are integers with ad − bc = 1.By taking these integers as multiples of some integer N we obtain In practice, the relevant finite group is the quotient of these two infinite groups, Γ N = Γ/Γ(N ), as we consider the representations of fields under Γ N .In terms of the generators, Γ N corresponds to imposing The chiral superfields φ i now transform under Γ N as weighted representations [4], where k is the modular weight, and ρ is a unitary representation of Γ N .
A notable difference of modular symmetries is that the would-be coefficients can be functions of τ and this then allows for these functions to transform as multiples under Γ N -the modular forms: An invariant term of the superpotential is then written as which is invariant as long as the tensor product of the Γ N representations contains a singlet (i.e., , and the weights cancel out, such that the term is unaffected by a modular action of γ. 2Generalizing the framework to include multiple modular symmetries is relatively straightforward in that one adds a modulus τ J for each of the modular symmetries Γ J N for J = 1, ..., M (in general the N can be different for each J, but we don't consider it here so we will keep the notation simpler).
A chiral superfield φ i now transforms under the multiple (commutative) symmetries as The modular forms are likewise generalized: Another aspect of (multiple) modular symmetry models is that the modular weights can be used to employ the weighton mechanism [13,14].By adding a weighton field which transforms typically as a singlet under the modular symmetry, but carries a non-vanishing modular weight, it is possible to arrange terms to be invariant with specific powers of this weighton field, thus suppressing the respective term, in a manner similar to the Froggatt-Nielsen mechanism [51] -but without having to introduce extra symmetry.

SU (5) Embedding
We are extending the Littlest Modular Seesaw to a grand unified setting, and a straightforward possibility is to extend the gauge symmetry to a SU (5) framework.We briefly review some SU (5) details and set our conventions.Further details about Grand Unified Theories can be found e.g. in [52,53].We furnish a 10 and a 5 SU (5) representations with the usual SM fields (including singlet heavy neutrinos) as follows: where the 10 is an anti-symmetric representation of SU (5), and so we omit the lower entries.
The relevant tensor products for the Yukawa terms are Y D : where Y u,d are the quark Yukawa matrices, and Y ℓ and Y D are the charged-lepton and Dirac neutrino mass matrices, respectively.We see that we must include scalars in a 5 representation to have a non-zero Y D , which automatically also leads to a non-zero Y ℓ and Y d .A minimal choice which provides a non-zero Y u is to include a scalar in a 5 representation.As we can see, these gauge assignments (required for the low-energy theory to be SM-like), relate the charged-leptons and down quarks, placing them in a single SU (5) multiplet.More specifically, if we include only a 5 scalar field, responsible for the Yukawas for the (low-energy) charged-leptons and down quarks, we unavoidably find (at the UV scale) This simple relation is not viable, and can be relaxed by the inclusion of a second scalar multiplet.Introducing a 45, provides a splitting between Y ℓ and Y d [54].With the inclusion of these two multiplets, the mass matrices are given by: relations which can be inverted to yield As a consequence, we see that the Yukawa matrices for the down quarks and charged-leptons become general.Nonetheless, we also see that texture zeroes are shared by both matrices, up to transposition.Thus, even though we include a second scalar multiplet to avoid the stringent relation Y d = Y T ℓ , the connection between charged-leptons and down quarks lingers on (a situation we will denote as Y ℓ ∼ Y T d during the rest of the paper).A second consequence of the choice of SU (5) as a gauge symmetry comes from the terms responsible for the up quarks Yukawa matrix.Since T contains both the LH and RH quarks, the Yukawa terms are given by T i H 5 T j , and are thus necessarily symmetric.
In summary, for our purposes here, in order to convert the Littlest Modular Seesaw into SU (5) grand unification, we need to consider non-trivial constraints on the Yukawa couplings.As had already been mentioned above, we will have symmetrical up quark Yukawa couplings and introduce additional scalars to make the Yukawa couplings of the charged lepton and down quarks viable.However, the modular symmetries will lead to texture zeroes in the charged-lepton mass matrices, which are preserved up to transposition, retaining consequences of the SU (5) unification.

The Model
The littlest modular seesaw model is a simple implementation of multiple modular symmetries that economically explains the leptonic sector flavour observables.The inclusion of an symmetry based explanation for the quark observables is a desirable next step.One interesting possibility is to take advantage of the SU (5) link between charged-leptons and down quarks.The inclusion of a 45 decouples this connection, except that it retains the symmetry protected zeroes in the mass matrices.We leverage this fact to design a model in which the symmetries still safeguard the littlest modular seesaw against large contributions from the leptonic sector, while enhancing the contribution of the down sector to the quark mixing.To retain the successes of the CSD(1 ± √ 6) lepton mixing predictions, the structure for Y ℓ cannot have large deviations from the diagonal shape.On the other hand, given the connection between the lepton and down-quark sectors (Y d ∼ Y T ℓ ), we need to be careful not to suppress the Y d contribution to the quark mixing.To this end, our goal is to employ lower and upper triangular Yukawa matrices for the charged-lepton and down-quark sectors, respectively.The point of the triangular shape for Y ℓ is to suppress the corrections to the lepton mixing (compared to the diagonal structure in the unspoiled CSD(n) framework).This suppression can be understood through a simplistic illustration of the 2 × 2 case: taking the matrix we compute the Hermitian matrices (assuming real Yukawas, for simplicity) The ensuing rotation angles are, assuming Clearly, we see that in the lower triangular case, the mixing will be suppressed due to hierarchical nature of the fermion masses, whereas the mixing in the upper triangular case can be O(1). 4 The UV nature of the model requires the running of the measured masses to some high scale.Here, we make use of the values shown in [55], obtained from [56,57], for tan β = 5 and a GUT scale at 2 × 10 16 GeV: For the neutrino observables, we use the NuFit 5.2 IR values [58,59]. 6aking the Cabibbo angle as a measure, λ ∼ 0.227, the experimental values are approximately 3 In this work, we follow the left-right convention for the Yukawa matrices. 4Due to the RH rotation freedom in the SM, we can more accurately describe the lower and upper triangular forms through their hermitian combinations together with m22 ≫ m11, rather than their unphysical Yukawa shapes.Regardless, we feel no confusion will arise throughout the paper.
We wish to have O(1) coefficients controlling the fermionic masses and mixings.Consequently, we can exploit the smallness of the quark mixing angles and naively use Eq. ( 15) to populate the entries of the Yukawa matrices, such that these are, by design, suppressed in such a way that both the quark mass hierarchy and the CKM matrix come out naturally: where the off-diagonal non-zero entries of Y ℓ are undetermined and denoted as "−", since the lower triangular shape suppresses the contributions to the leptonic mixing, and the main driver behind the PMNS mixing matrix comes from the Dirac neutrino structure, in the modular CSD(1 ± √ 6) set-up.As for the neutrino sector, due to the built-in suppression mechanism in the form of the Type-I seesaw, we do not require any specific suppressions.We stress that the matrices shown in Eq. ( 19) are derived merely from the experimental values, and are not necessarily attainable in a specific set-up.Indeed, the SU (5) gauge symmetry, assuming the same order of magnitude for all Yukawas, will forbid different suppressions in Y ℓ and Y T d , as we will see later.Our model relies on a SU (5) gauge symmetry, supplemented by 3 distinct S 4 modular symmetries.The assignments of the fields are given in Table 1, both for the gauge and modular symmetries.We note that although we use non-integer weights for the fields, only even-weighted Yukawa modular forms are considered, consistent with the requirement of invariance under the S 4 modular group.Indeed, rational modular weights for the fields are also obtained from top-down constructions in [61][62][63].As such, the present framework continues to be that of modular invariance, and not that of metaplectic models [64,65], since we do not consider half-integer modular forms.
Motivated by the modular CSD(1 ± √ 6) structure, we do not take the values for the moduli as free parameters, and set them to the relevant stabilisers [7,8].As such, we take where the choice of Eq. (20a) corresponds to the CSD(1 + √ 6) case, whereas Eq. (20b) gives rise to CSD(1 − √ 6).In the basis of [7,22], these correspond to more familiar fixed points: We emphasise that the fixed points in Eq.20c arise from the well known single modulus fixed points τ = i and τ = ω.With three moduli, the two fixed points i and i+ 2 are simply related but inequivalent.The superpotential responsible for the up-quark mass matrix comes from T i H 5 T j couplings: Field SU (5) Table 1: Assignments of the fields under the SU (5) gauge symmetry and the representations and weights under the 3 modular symmetries (S A 4 , S B 4 , S C 4 ) considered.We omit fields that are necessary for a consistent UV completion, such as messenger fields to complete the non-renormalizable terms, as well as the driving fields responsible for the bi-triplet and weighton VEVs.
where we suppress Y (0) 1 (τ A,B ) from the notation, and Λ stands for the relevant UV scale for a particular non-renormalizable operator, which we take to be universal for notational convenience.The contributions to Y d (and similarly for Y ℓ ) come from the couplings to both H 5 and H 45 .The superpotential will read the same for both scalars, with H 5 and H 45 exchanged, and different Yukawa couplings: + 5 → 45 .
The relevant modular forms (for τ C = ω) are given by Y (2) and thus, the resulting Yukawa matrices are The superpotentials of Eqs. ( 21) and ( 22) responsible for the quark and charged-lepton masses rely on non-renormalizable operators, through multiple weighton insertions.After the weighton fields acquire a non-zero VEV, ǫ F,T = φ F,T /Λ, the quarks and charged-leptons get contributions to their masses a la Froggatt-Nielsen, as per the weighton mechanism.If we assume O(1) coefficients, together with ǫ F ∼ λ 3 , and ǫ T ∼ λ 2 , we see that the mass matrices are close to those of Eq. ( 19): where the up quark Yukawa matrix is symmetric.We see that we are unable to get different suppressions for Y ℓ and Y d , due to the SU (5) nature of the model.Nonetheless, if we identify ǫ T ∼ λ 2 and ǫ F ∼ λ 3 , all the entries would have the desired suppressions, except for the (1, 2) entries of both Y u and Y d carry an extra suppression of λ, and so does the (1, 1) entry of Y d (yielding, however, the correct suppression for Y ℓ ).As such, we expect from the start that the model is able to fit the quark masses and mixings, as well as the charged-lepton masses, with O(1) coefficients, as the model is designed such that the weighton insertions could be responsible for most of the observed hierarchies.
We turn now to the neutrino sector.The neutrino Dirac Yukawa matrix comes from F H 5 N c couplings.Due to the requirement of invariance under the modular symmetries, we see that Y D can only be non-zero via couplings to the bi-triplets Φ AC and Φ BC .The allowed superpotential for the neutrino Dirac Yukawa matrix reads Given the moduli of Eq. ( 20), the relevant Yukawa modular forms are After both the bi-triplets and the weighton acquire a non-zero VEV, the terms of Eq. ( 26) populate the neutrino Dirac Yukawa matrix as Lastly, we analyse the relevant terms for the RH neutrino mass matrix.The modular assignments of Table 1 do not allow for the presence of bare mass terms, otherwise allowed by gauge invariance.However, similarly for the remaining fermions, we can build non-renormalizable terms which will generate mass terms for the heavy neutrinos below an appropriately large scale.The relevant Yukawa modular forms here will transform as 1 ( ′ ) and, as it happened for the cusp, there are vanishing modular forms at τ A and τ B .We find that, up to weight 10, the relevant non-zero modular forms are The Majorana superpotential is given by where, as usual, we omit any modular form of weight 0. Any mixed term is forbidden by the modular symmetries.The ensuing Majorana mass matrix is then given by Through the type-I seesaw mechanism, the effective mass matrix for the light neutrinos becomes with n = 1 ± √ 6, as desired for the CSD(1 ± √ 6) predictions.This can be written in more compact notation as where are real parameters and β is an undetermined phase.This shows that the neutrino mass matrix is completely determined by three real parameters m a , m b and β, where n = 1 ± √ 6, making it a highly predictive scheme, which successfully describes the current data as recently discussed [66].
We note as a final remark that, due to the choice of assignments of Table 1, the superpotentials of Eqs. ( 21), ( 22), (26), and (30) do not have higher order corrections stemming for further insertions of weightons.A more technical note is shown in Appendix B, where we highlight the importance of having only one way to generate the couplings of Eq (26).
We expect the model to be compatible with experiment, due to its design.For completeness, we show here one point to showcase that indeed we can get the UV values for the quark masses and mixings with O(1) coefficients, and negligible χ 2 : 24i , y uc = 0.97294 e −2.59i , y ′ uc = 0.93204 e 0.0393i , y ut = 0.97272 e 1.20i , y cc = 1.0264 e 1.53i , y ct = 0.92436 e −0.461i , y tt = 0.53034 e −2.34i , y dd = 2.6549 e 1.10i , y ds = 2.1282 e 0.555i , y db = 1.0022 e −1.07i , y ss = 0.62888 e −2.97i , y sb = 0.93386 e 2.76i , y bb = 0.56589 e 1.10i , As expected, we see that ǫ F ∼ λ 3 , and ǫ T ∼ λ 2 .Moreover, we see that we can easily fit the quark masses and mixings with O(1) Yukawas, due to having the correct suppressions on the Yukawa matrices.As mentioned earlier, the (1, 1) and (1, 2) entries of Y d have an extra λ suppression comparing to our naïve guess.In that sense, the associated couplings (y dd and y ds ) need to be larger to compensate.That is clearly seen in the numerical result of Eq. ( 34), in the relative hierarchy between y dd,ds and the remaining Yukawas.Turning now to the neutrino and charged-leptons, we provide a benchmark for using the results obtained for the quarks, namely, taking ǫ T and ǫ F from Eq. (34).We find, for the n = 1 + √ 6 case, and taking into account the SK-atmospheric data: y ee = 1.0740e 2.48i , y µe = 1.0000e 2.62i , y τ e = 1.0000e 0.495i , y µµ = 2.9497e −0.389i , y τ µ = 4.0000e 0.490i , y τ τ = 0.54344e 1.31i , which is very close to the LMS best-fit point, with χ 2 ∼ 1.9.This good fit to the flavour observables in a SU (5) unified model is obtained through a combination of Georgi-Jarlskog factors, the upper / lower diagonal form for the matrices of the down quarks and charged leptons respectively, and the weighton mechanism, and demonstrate clearly some of the advantages of employing (multiple) modular symmetries in theories of flavour.

Conclusion
A grand unified theory of flavour is a desirable goal, but not easy to achieve.The connections between families imposed by unification restrict the solutions to the flavour problem.In this paper we tackled the challenge by embedding into SU (5) unification the littlest modular seesaw model.This is a model that describes the lepton sector extremely well through multiple modular flavour symmetries.We surmounted the typical difficulty arising from the relation between charged leptons and down quarks by employing Georgi-Jarlskog factors arising from appropriate SU (5) multiplets, and, from the modular flavour symmetry, constructing an upper-diagonal matrix in the flavour symmetry basis for the down quarks.Due to this, the contributions to quark mixing are sizeable at the same time that the transposed charged lepton matrix is lower-diagonal, such that its off-diagonal entry contributions to the leptonic mixing are suppressed by the hierarchical charged lepton mass ratios.We employ also two weightons, which enable us to justify the hierarchical entries in the mass matrices through the use of the modular wieghts, a mechanism which is reminiscent of typical Froggatt-Nielsen, but without requiring the introduction of an extra symmetry.In conclusion, we present an elegant SU (5) unified, multiple modular flavour symmetry model which accounts for the flavour observables in a unified setting, with an emphasis on the predictivity of the leptonic sector, and the use of weightons that explain all the fermion mass hierarchies.
A S 4 Group Theory and Modular Forms at the Cusp The generators of S 4 obey We follow the S 4 basis of [46], where the representation matrices are The tensor products are given by where we only show the relevant Clebsch-Gordan coefficients for our model, and the remaining can be found in [46]: Using this basis, the relevant fixed points for our model are As for the modular forms we have, at the lowest weight in S 4 , Y Higher weight modular forms are obtained through the tensor products of lower-weight modular forms: which can be easily computed for the cusp, and we see that, up to weight 12: where we only show the non-vanishing modular forms.It is clear that the pattern repeats, such that the non-vanishing modular forms at k = 2, 4, 6 and k = 8, 10, 12 are identical, respectively.Additionally, given Eq. ( 42), and that the modular forms of weight 6 and 12 are identical, then there is no difference in computing Y (2) ⊗ Y (6) and Y (2) ⊗ Y (12) .Thus, at the cusp, it becomes obvious that the modular forms are given by: assuming they exist in at a certain modular weight.This is useful for model building at fixed points, since it allows us to easily identify the shape of the modular forms of higher weights, without the need for actual computation.

B Possible Corrections to the CSD(n) Matrices
At first glance, the assignments of the model shown in the main text appear more convoluted than necessary.Indeed, it is possible to find a seemingly simpler model, which includes all the terms and invariants we find in the main text.However, it is important to check if there are (non-negligible) corrections to any of the structures found for the relevant Yukawa matrices.In this Appendix, we show a simple example of this.Table 2 shows one possible set of assignments for the fields that also lead to the structures found in the main text for Y u and Y d .Namely, we find as we do with the model in the main text.Moreover, the charges of the weightons (φ F , φ T ) under S A 4 will forbid higher-order corrections to these matrices.
As we can see from the weight assignments for the RH neutrinos, here the SU (5) singlets will have bare mass terms, at the renormalizable level, contrary to what we see in Eq. (30).Mixed terms are still forbidden by the absence of the Y (2) 1 modular form, needed to make a N c A N c B term invariant.The last ingredient needed is to reproduce the Dirac mass matrix compatible with the CSD(n) structure.There, we find that the invariants are present, as required, similarly to Eq. ( 26).On the other hand, we can now make another set of invariants, by replacing φ T with φ F .If we follow the same tensor product contractions as the terms in w D , these would be forbidden by the absence of the Y (2) 1 modular form.However, we can contract the term as (we show the example for F N c A )

Y
(2) which spoil the Y D structure.At first glance, we could argue that we could choose the model's messengers such that the terms in Eq. ( 46) are present, but those of Eq. ( 47) are absent.However, we checked that, for the simplest choice of messengers, both terms are necessarily present.This is shown in the diagrams of Fig. 1, where we can clearly see that the same set of messengers lead to the existence of A seemingly simpler assignment under the three modular symmetries, but which lead to nonnegligible contributions which spoil the CSD(n) structure.As in the main text, we do not show the messenger fields nor any necessary driving fields.both terms.The assignments of the model presented in the main text are mostly motivated to eliminate these terms, and keep an unspoiled Y D , such that the corrections to the CSD(n) structure arise solely from the non-diagonal Y ℓ , in a suppressed manner.

Figure 1 :
Figure 1: The diagrams leading to the desired (top) and undesired (bottom) terms for w D .The modular forms refer to S A 4 , as the remaining are trivial.The messengers are fermionic SU (5) singlets, represented by the S A 4 and S C 4 representations and weights, and assumed to transform trivially under S B 4 .