SU(5) Grand Unified Theory with A4 Modular Symmetry

PHYSICAL REVIEW D 101, 015028 (2020)

SU(5) grand unified theory with A4 modular symmetry

† ‡ Francisco J. de Anda ,2,* Stephen F. King,1, and Elena Perdomo1, 1School of Physics and Astronomy, University of Southampton, SO17 1BJ Southampton, United Kingdom 2Tepatitlán’s Institute for Theoretical Studies, C.P. 47600, Jalisco, M´exico

(Received 1 April 2019; published 30 January 2020)

We present the first example of a grand unified theory (GUT) with a modular symmetry interpreted as a family symmetry. The theory is based on supersymmetric SUð5Þ in 6d, where the two extra dimensions are compactified on a T2=Z2 orbifold. We have shown that, if there is a finite modular symmetry, then it can i2π=3 only be A4 with an (infinite) discrete choice of moduli, where we focus on τ ¼ ω ¼ e , the unique solution with jτj¼1. The fields on the branes respect a generalized CP and flavor symmetry A4 ⋉ Z2 which is isomorphic to S4 which leads to an effective μ − τ reflection symmetry at low energies, implying maximal atmospheric mixing and maximal leptonic CP violation. We construct an explicit model along these lines with two triplet flavons in the bulk, whose vacuum alignments are determined by orbifold boundary conditions, analogous to those used for SUð5Þ breaking with doublet-triplet splitting. There are two right-handed neutrinos on the branes whose Yukawa couplings are determined by modular weights. The charged lepton and down-type quarks have diagonal and hierarchical Yukawa matrices, with quark mixing due to a hierarchical up-quark Yukawa matrix.

DOI: 10.1103/PhysRevD.101.015028

I. INTRODUCTION of the discrete non-Abelian group symmetry such as A4 and S4 which may be identified as a remnant symmetry of the The flavor puzzle, the question of the origin of the three extended Poincar´e group after orbifolding. families of quarks and leptons together with their curious Some time ago it was suggested that modular symmetry, pattern of masses and mixings, remains one of the most when interpreted as a family symmetry, might help us to important unresolved problems of the Standard Model provide a possible explanation for the neutrino mass (SM). Following the discovery of neutrino mass and mixing, matrices [13,14]. Recently it has been suggested that whose origin is fundamentally unknown, there are now neutrino masses might be modular forms [15], with con- almost 30 undetermined parameters in the SM, far too many straints on the Yukawa couplings. This has led to a revival for any complete theory. The lepton sector in particular of the idea that modular symmetries are symmetries of the involves large mixing angles that suggest an explanation in extra-dimensional spacetime with Yukawa couplings deter- terms of discrete non-Abelian family symmetry [1,2]. mined by their modular weights [16]. However to date, no Furthermore, such discrete non-Abelian family symmetries attempt has been made to combine this idea with orbifold have been combined with grand unified theories (GUTs) in GUTs in order to provide a unified framework for quark order to provide a complete description of all quark and and lepton masses and mixings. lepton (including neutrino) masses and mixings [3]. In this paper we present the first example in the literature It is well known that orbifold GUTs in extra dimensions of a GUT with a modular symmetry interpreted as a family (ED) can provide an elegant explanation of GUT breaking symmetry. The theory is based on supersymmetric SUð5Þ and Higgs doublet-triplet spitting [4]. Similarly, theories in 6d, where the two extra dimensions are compactified on involving GUTs and flavor symmetries have been formulated i2π=3 a T2=Z2 orbifold, with a twist angle of ω ¼ e . Such in ED [5–12]. These EDs can help us to understand the origin constructions suggest an underlying modular A4 symmetry with a discrete choice of moduli. This is one of the main *[email protected] differences of the present paper as compared to recent † [email protected] works with modular symmetries which regard the modulus ‡ [email protected] τ as a free phenomenological parameter [15,16].We construct a detailed model along these lines where the Published by the American Physical Society under the terms of fields on the branes are assumed to respect a flavor and the Creative Commons Attribution 4.0 International license. CP A ⋉ Z Further distribution of this work must maintain attribution to generalized symmetry 4 2 which leads to an the author(s) and the published article’s title, journal citation, effective μ − τ reflection symmetry at low energies, imply- and DOI. Funded by SCOAP3. ing maximal atmospheric mixing and maximal leptonic CP

2470-0010=2020=101(1)=015028(17) 015028-1 Published by the American Physical Society DE ANDA, KING, and PERDOMO PHYS. REV. D 101, 015028 (2020) 0 violation. The model introduces two triplet flavons in the ω1 ω ab ω1 → 1 ; bulk, whose vacuum alignments are determined by orbifold 0 ¼ ω2 ω2 cd ω2 boundary conditions, analogous to those used for SU 5 ð Þ ab breaking with doublet-triplet splitting. There are also two where ∈ SLð2; ZÞð2Þ right-handed neutrinos on the branes whose Yukawa cd couplings are determined by modular weights. The charged lepton and down-type quarks have diagonal and hierarchi- or equivalently if a; b; c; d ∈ Z and ad − bc ¼ 1. These are cal Yukawa matrices, with quark mixing due to a hierar- called modular transformations and form the modular chical up-quark Yukawa matrix. group Γ [15]. Without loss of generality, the lattice vectors The remainder of the paper is organized as follows. In may be rescaled as 2 Sec. II we discuss the orbifold T =Z2 and its symmetries, as follows. In Sec. II A, we give a review of modular trans- ω1 1 1 → ¼ ; ð3Þ formations while in Sec. II B we describe how the orbifold ω2 τ ω2=ω1 T2=Z2 is only consistent with modular A4 symmetry and a choice of modulus. In Sec. II C, we explicitly show the such that the torus is equivalent to one whose periods T =Z ω ei2π=3 A orbifold 2 2 with twist angle ¼ and modular 4 are 1 and τ ¼ ω2=ω1 and we can restrict τ to the upper half- symmetry. In Sec. II D we study the remnant symmetry after plane H ¼ Imτ > 0. The modular transformations on the compactification on the T2=Z2 orbifold, while Sec. II E rescaled basis vectors which leave the lattice invariant are connects this remnant symmetry and the modular symmetry. given by1 In Sec. II F we discuss the enhanced A4 ⋉ Z2 on the branes. In Sec. III, we present the field content of the SUð5Þ GUT 1 1 ω0 aτ b → ; τ0 2 þ : with A4 modular symmetry and a Uð1Þ shaping symmetry, where ¼ ¼ ð4Þ τ τ0 ω0 cτ þ d including the Yukawa sector and the specific structure for the 1 effective alignments that the modular symmetry can gen- A SLð2; ZÞ transformation on the modulus parameter τ and erate, resulting in the low energy form of the SM fermion its negative are equivalent, as can be seen from Eqs. (2) and mass matrices which we show can lead to a very good fit to (4). Therefore, we can use the infinite discrete group the observables. Finally in Sec. IV we present our con- PSLð2; ZÞ¼SLð2; ZÞ=Z2, generated by clusions. In order to make the paper self-contained, some necessary background information is included in the S∶ τ → −1=τ T∶ τ → τ 1; Appendixes. In Appendix A we show the explicit proof and þ ð5Þ that only the A4 modular symmetry is consistent with the branes, with specific choices of modulus. We supplement to describe the transformations that relate equivalent Γ¯ the general A4 group theory in Appendix B, the consistency tori. This is also called the modular group satisfying Γ¯ Γ= 1 2 conditions for generalized CP symmetry consistent with A4 ¼ f g. The generators of the infinite-dimensional in Appendix C and the general theory for modular forms in modular group can also be written as Appendix D. Finally we show sample fits of the observed data in Appendix E. 01 10 S ¼ ;T¼ : ð6Þ −10 11 II. ORBIFOLDING AND SYMMETRIES They satisfy the presentation A. Review of modular transformations In this subsection we present the general theory of modu- Γ¯ ≃ fS; TjS2 ¼ðSTÞ3 ¼ Ig=f1g; ð7Þ lar transformations. The structure of the extra-dimensional torus is defined by the structure of the lattice by where S; T ∈ SLð2; ZÞ. We will be considering the finite-dimensional discrete M z ¼ z þ ω1;z¼ z þ ω2; ð1Þ subgroups by imposing an additional constraint on T , where M is a positive integer, where ω1;2 are the lattice basis vectors. The variable z z x ix x ¯ 2 3 M refers to the complex coordinate ¼ 5 þ 6, where 5 ΓM ≃ fS; TjS ¼ðSTÞ ¼ T ¼ Ig=f1g; ð8Þ and x6 are the two extra-dimension coordinates. The torus is then characterized by the complex plane C modulo Λ Λ 1Where we have relabeled the integers a → d, b → c, c → b, a two-dimensional lattice ðω1;ω2Þ, where ðω1;ω2Þ ¼ d → a to be consistent with conventional notation. fmω1 þ nω2;m;n∈ Zg, i.e., T2 ¼ C=Λ ω ;ω . The lattice 2 ð 1 2Þ The modular group Γ refers to SLð2; ZÞ, while Γ¯ is used for is left invariant under a change in lattice basis vectors PSLð2; ZÞ and takes into account the equivalence of an SLð2; ZÞ described by the general transformations matrix and its negative.

015028-2 SUð5Þ GRAND UNIFIED THEORY WITH A4 MODULAR … PHYS. REV. D 101, 015028 (2020) where S; T ∈ SLð2; ZMÞ. These groups, with M ≤ 5, are This paper’s approach is to focus on the orbifold first and ¯ ¯ isomorphic to the known discrete groups as Γ2 ≃S3, Γ3 ≃A4, then derive the modular symmetries, instead of going ¯ ¯ Γ4 ≃ S4, Γ5 ≃ A5. directly into the modular symmetries. We will restrict We now introduce a convenient (if nonunique) repre- ourselves to the case where jω1j¼jω2j, which happens sentation for the modular transformations consistent with when p ¼ −1, q ¼ −1. This way we focus on studying the the presentation in Eq. (8), effects only of the angle between both vectors. We can, without loss of generality, choose ω1 ¼ 1. Furthermore, the 01 e−2iπ=M 0 modular symmetries require τ to lie in the upper complex S ¼ ;TM ¼ ; ð9Þ −10 ð Þ 1 e2iπ=M plane; in this case the only solutions to Eqs. (12) and (13) i2π=3 are ω2 ¼ ω ¼ e . This uniquely fixes the modulus Γ¯ 2 which satisfies the presentation of the M group, for any coming from the orbifold T =Z2. We emphasize that this integer M>2. This representation will be useful in the is one of the main differences of the present paper as following discussion. compared to recent works with modular symmetries which regard the modulus τ as a free phenomenological parameter 2 T2=Z B. Why the orbifold T =Z2 suggests modular A4 [15,16]. In our work, we assume a specific orbifold 2, symmetry with modulus τ = ω for which we have shown that one consistent choice for a A τ In this subsection we present an argument which shows surviving modular symmetry is 4 with fixed modulus , 2 although we shall not address the problem of moduli that a particular T =Z2 orbifold (as assumed in this paper) stabilization [17]. suggests an underlying modular A4 symmetry with specific modulus parameters. 2 T2=Z ω ei2π=3 We begin by defining the orbifold T =Z2 in terms of two C. The orbifold 2 with = A arbitrary lattice vectors ω1 and ω2, and modular 4 symmetry Following the argument of the previous subsection, we z z ω ;zz ω ;z−z: 2 ¼ þ 1 ¼ þ 2 ¼ ð10Þ henceforth focus on the orbifold T =Z2 with particular twist angle denoted as ω ¼ ei2π=3, identified as the modulus The action of the orbifold in Eq. (10) leaves four invariant 3 τ associated with a particular finite modular symmetry A4, 4d branes given by A where 4 is the only choice consistent with this orbifold. ω ω ω ω This orbifold then corresponds to the identification z¯ 0; 1 ; 2 ; 1 þ 2 : ¼ 2 2 2 ð11Þ z ¼ z þ 1;z¼ z þ ω;z¼ −z; ð14Þ After compactification, the symmetries of the branes where the first two equations are the periodic conditions remain unbroken; therefore it is relevant to study any from the torus T2 and the third one is the action generated possible symmetry among the branes which will affect the by the orbifolding symmetry Z2. The twist corresponds to fields localized on them. Therefore, we want to check if the ω ¼ ei2π=3. The orbifold symmetry transformations leave modular transformations in Eq. (9) leave an invariant set of M four invariant 4d branes as shown in Fig. 1: branes for some value of . At this stage the modulus τ ¼ ω2=ω1 can apparently take any value. However we 1 ω 1 þ ω z¯ ¼ 0; ; ; : ð15Þ present a proof in Appendix A that only the A4 symmetry is 2 2 2 consistent, meaning that M ¼ 3, when the basis vectors are related by The transformations

ei2π=3 þ 2p þ 2 S∶ z → z þ 1=2 or z → z þ ω=2; ω2 ¼ − ω1: ð12Þ 2q þ 1 T∶ z → ω2z; The p and q are integers satisfying that U∶ z → z or z → −z; ð16Þ

ð2p þ 1Þðp þ 1Þþq þ 1 permute the branes and leave invariant the set of four branes r ¼ ð13Þ 2q þ 1 in Eq. (15). These transformations satisfy is an integer, which has infinitely many discrete solutions. S2 ¼ T3 ¼ðSTÞ3 ¼ 1; Furthermore, since the modular forms restrict τ to be in the U2 ¼ðSUÞ2 ¼ðTUÞ2 ¼ðSTUÞ4 ¼ 1; ð17Þ upper complex plane, then q<0. where the first line is the presentation of the group A4 and 3 The notation for the lattice vectors ω1;2 should not be both lines complete the presentation of S4 [1]. In Fig. 1 we confused with the twist angle ω ¼ ei2π=3. show how these transformations act on the extra-

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(a) (b)

FIG. 1. Visualization on the remnant A4 symmetry after orbifolding.

“ A ” S; T ∈ SL 2; Z dimensional space and how the remnant 4 symmetry is where ð3Þ ð 3Þ, so that the branes are indeed ¯ realized. invariant under the discrete modular group Γ3 ≃ A4.Aswe Fixing M ¼ 3, the set of branes is invariant under the will see in Sec. II F, this symmetry will be enlarged. modular transformations 01 ω2 0 D. Remnant brane symmetry T2=Z ω ei2π=3 S ¼ ;T3 ¼ ; ð18Þ for 2 with = −10 ð Þ 1 ω So far we have shown that the choice of orbifold T T2=Z on the lattice vectors ð1; ωÞ . These transform the basis 2 is consistent with the finite modular symmetry vectors as A4 with a discrete choice of moduli, where we focus on i2π=3 τ ¼ ω ¼ e . Now we will take a step back, forget about 1 ω 1 ω2 −1 − ω S ;T ; modular symmetries for a while, and just consider the ¼ ð3Þ ¼ ¼ i2π=3 ω −1 ω 1 þ ω2 −ω symmetries of the branes with a twist angle ω ¼ e .We will discover an S4 symmetry that has apparently nothing to ð19Þ do with modular symmetry, which we refer to as “remnant 2 S4 symmetry.” In the next subsection we shall show how (noting that 1 þ ω þ ω ¼ 0) leaving the lattice invariant as the subgroup remnant A4 symmetry is related to the can be seen from Fig. 2. previous A4 finite modular symmetry. The matrices S; T 3 fulfill the presentation of the group ð Þ In this section, we will find that the branes are invariant they generate to be under an S4 and its subgroup A4 symmetry which can be 2 3 3 ¯ identified as a remnant symmetry of the spacetime sym- fS;T 3 jS ¼ T ¼ðST 3 Þ ¼ Ig=f1g¼Γ3 ≃ A4; ð20Þ ð Þ ð3Þ ð Þ metry after it is broken down to the 4d Poincar´e symmetry

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FIG. 2. Visualization of the lattice invariant transformations, S and T. through orbifold compactification. Here, we assume that transformation. This way we may identify the remnant the spacetime symmetry before compactification is a 6d A4 symmetry of the branes as a modular symmetry since the Poincaré symmetry. The compactification breaks part of effect on the branes of each type of transformation is this symmetry. However, due to the geometry of our identical; it is just a choice of “picture” (active or passive) orbifold with twist angle ω ¼ ei2π=3, a discrete subgroup which we choose. is left unbroken. This group may be generated by the The action of the modular symmetry on the branes spacetime transformations (which belong to the extra- behaves as a “normal” symmetry (i.e., modular forms are dimensional part of the 6d Poincaré). not relevant) since fields located on the brane do not depend The orbifolding leaves four invariant branes, and this on the extra-dimensional coordinate. The modular sym- specific orbifold structure leaves them related by the group metry can therefore be imposed as any usual symmetry. In S4. This symmetry, together with 4d Poincaré transforma- the orbifold T2=Z2, the branes are only consistent with the ¯ tions, is a subgroup of the extra-dimensional Poincaré modular group Γ3, as shown in Appendix A. Any theory symmetry that survives compactification. This is the with this orbifold and fields allocated on the branes will ¯ standard “remnant symmetry” [6,18]. only be consistent with the modular symmetry Γ3. In such a Any field located in the branes will transform under the setup, the branes see the finite modular symmetry as simply 4d Poincaré group as usual. Since the branes transform into equivalent to a remnant symmetry, a subgroup of the extra- each other by the remnant symmetries, the fields on the dimensional Poincaré group. brane should also transform under them. The four branes We can see in Eq. (18) that the S, T transformations (and ¯ transform under the remnant S4 symmetry and we choose therefore the Γ3 modular transformations) correspond to the embedding of the representation 4 → 3 þ 1 so that the specific passive reflections, rotations and translations. In ¯ fields in the branes can only transform under those this way the Γ3 must be a subgroup of the 6d Poincaré group. irreducible representations [7,11]. The fields in the branes All modular groups are. However not all modular groups are do not depend on z but are permuted into each other by the consistent with the invariant branes, as we have shown. S, T, U transformations. On the other hand, fields in the bulk, which feel the extra dimensions, will also transform under some repre- A sentation of the 6d Poincaré; however in this case they will E. The connection between remnant 4 symmetry ¯ transform under a nonlinear realization of this Γ3 sym- and finite modular A4 symmetry metry, and this is precisely what are referred to as the We have shown that the set of branes is invariant under a modular forms [15]. remnant S4 (and its subgroup A4) subgroup of the extra- Γ¯ S We conclude that the modular symmetry 3 acts either as dimensional Poincaré symmetry. We shall return to the 4 a linear or nonlinear realization of the remnant symmetry symmetry in the next subsection and we now show that the A4, depending on whether we are concerned with brane A remnant 4 symmetry can be identified with the finite fields or bulk fields. modular A4 symmetry discussed earlier. Essentially, if we impose a modular symmetry A4 on the whole space, its A ⋉ Z action on the branes is the same action as the remnant F. Enhanced 4 2 symmetry of the branes spacetime symmetry; i.e., it permutes the branes but leaves We now recall that, in our setup, the brane fields enjoy a invariant the whole set. The modular symmetry acts on larger S4 symmetry than the remnant A4 symmetry, as the basis vectors of the torus while the remnant symmetry is shown in Sec. II D. However this larger S4 symmetry is not a spacetime symmetry and acts on the fixed points; related to a finite modular symmetry, since the branes can therefore from the point of view of the branes, the “remnant only be invariant under the modular transformations ¯ A4 symmetry” is an active transformation while the corresponding to Γ3 ≃ A4, and is not enjoyed by the fields finite modular A4 symmetry is an equivalent passive in the bulk.

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We note here that S4 ≃ A4 ⋉ Z2. The symmetry gen- The A4 modular symmetry will require the Yukawa erated by U from Eq. (16) is a remnant symmetry of the couplings to be specific modular forms, while the Z2 orbifolding process, but it cannot be interpreted as a symmetry will further restrict the possible mass matrix modular transformation. We conclude that the remnant structure so that the theory has strong predictions for ¯ symmetry of the branes is Γ3 ⋉ Z2 ≃ A4 ⋉ Z2. The Z2 leptons [20]. As we shall see later, the up quarks will lie symmetry is generated by C · U where U is the usual matrix in different A4 singlets with modular weight zero, so that representation of the generator from S4 and C stands for only the subgroup Z3 is a remnant while the Z2 behaves complex conjugation of the complex coordinate, which is trivially. This forces stringent relations for the lepton mass equivalent to a change of sign in x6, i.e., the parity matrices but not for the quarks. ψ transformation of the sixth dimension P6. The Z2 is not All the fields in the bulk will transform under the a modular symmetry while the A4 is. The product of both modular transformations symmetries is not direct since the generator U does not aτ b A S þ −k commute with all 4 generators and is the corresponding 4 τ → ; ψ → ðcτ þ dÞ ρψ; ð22Þ generator. cτ þ d After compactification, the remnant Z2 (which is not a where ρ is the usual matrix representation of the corre- subset of A4) acts on the brane fields generalized CP sponding A4 transformation. Each field has a weight −k, transformation where the transformations P1; …;P5 are ˜ with no constraint in k since the fields are not modular trivial while P6 ¼ P6U, where P6 is the trivial parity forms. The superfields that are located on the brane do not transformation, while the U is a family transformation depend on the extra dimensions and therefore they must [19]; thus a field transforms as have weight zero [15]. We arbitrarily choose a weight for each of the bulk fields. Piϕðx1; …;xi; …;x6Þ¼ϕðx1; …; −xi; …;x6Þ; The whole field content is listed in Tables I and II. The fields that do not have weight or parity under the boundary for i ¼ 1; …5; conditions are located on the branes and feel the symmetry P6ϕðx1; …;xi; …;x6Þ¼Uϕðx1; …;xi; …; −x6Þ: ð21Þ A4 ⋉ Z2; see Table I. The transformations of the fields

CP Under the fields transform as shown in Appendix C.As TABLE I. Fields on the branes, including matter and right- stated before, this effective symmetry transformation only handed neutrino superfields. A working set of charges is affects nontrivially the brane fields and the fields in the bulk fq1;q2;q3g¼f2; 0; 1g. Note that the 3 representations on the are unaffected, transforming under canonical CP and not brane transform under A4 ⋉ Z2 as shown in Table VI and forced to preserve it. Thus in our approach the generalized Eq. (C4). CP is a remnant symmetry in a particular sector of the Representation theory, corresponding to the fields on the branes. A ⋉ Z SU 5 U 1 We have shown that the remnant Z2 symmetry on the Field 4 2 ð Þ ð Þ CP ¯ branes behaves as an effective generalized transforma- F 3 5 q1 þ 2q3 c tion. In Appendix C we check its compatibility with the A4 Ns 11q1 c flavor symmetry and find that it is consistent, as indeed it Na 114q1 must be. ξ 11−2q1

III. SU(5) GUT WITH A4 MODULAR SYMMETRY TABLE II. Fields on the bulk used in constructing the model, including matter, Higgs and flavon superfields. A working set of A. The model charges is fq1;q2;q3g¼f2; 0; 1g. In this section we construct a supersymmetric SUð5Þ 2 i2π=3 Representation Localization GUT model on a 6d orbifold T =Z2 with twist ω ¼ e , A SU 5 U 1 P P P with an A4 modular symmetry as a flavor symmetry, Field 4 ð Þ ð Þ Weight 0 1=2 ω=2 Z 00 extended by the 2 symmetry on the branes. Furthermore T1 1 10 q3 þ 4q1 −γ þ1 1 1 U 1 0 we impose a global ð Þ as a shaping symmetry. We impose T2 1 10 q3 þ 2q1 −γ þ1 1 1 different boundary conditions at each invariant brane, which T3 110 q3 −γ þ1 1 1 break the original symmetry into the minimal supersym- H5 15 −2q3 −α þ1 þ1 þ1 U 1 0 ¯ metric standard model (MSSM). The ð Þ as a shaping H5¯ 1 5 q2 α þ γ þ1 þ1 þ1 symmetry forbids any higher order terms, while a discrete ϕ1 31−q2 − q1 − 3q3 −α þ1 þ1 þ1 ZN would allow them, the smaller the N, the corrections ϕ2 31 −3q1 α − β þ1 −1 þ1 would appear at lower order.

015028-6 SUð5Þ GRAND UNIFIED THEORY WITH A4 MODULAR … PHYS. REV. D 101, 015028 (2020) under this symmetry are discussed in Appendix B. The 3 P0 ¼ I3 × I5; representations on the brane transform under A4 ⋉ Z2 are P1=2 ¼ T1 × diagð−1; −1; −1; 1; 1Þ; as shown in Table VI and Eq. (C4). The field F contains the MSSM fields L and dR and is a Pω=2 ¼ T2 × diagð−1; −1; −1; 1; 1Þ; ð27Þ flavor triplet. It is located on the brane. The fields Ti contain the MSSM uR, eR, Q; they are three flavor singlets. where I3;T1;2 ∈ SUð3Þ, while I5; diagð−1; −1; −1; 1; 1Þ ∈ There are two copies of each T with different parities under SUð5Þ, and explicitly the boundary conditions; as we shall see in the next section, 0 1 0 1 this allows different masses for down quarks and charged 10 0 100 Nc B C B C leptons. There are only two right-handed neutrinos a;s. T1 ¼ @ 0 −10A;T2 ¼ @001A ¼ U; ð28Þ h H The MSSM Higgs fields u;d are inside the 5;5¯ respec- 00−1 010 tively. We have two flavons ϕ1;2 that help to give structure to the fermion masses. Finally, the field ξ generates the and the last boundary condition is fixed and defined by the hierarchy between the masses ala` Froggat-Nielsen [21]. P P P P P P other boundary conditions as ð1þωÞ=2 ¼ 0 1=2 0 ω=2 0 ¼ T1T2 ×I5. B. GUT and flavor breaking by orbifolding The boundary condition P0 breaks the effective extended N 2 → N 1 SUSY. The boundary condi- Since the orbifold has the symmetry transformations of ¼ ¼ tions P1=2;ω=2 leave their corresponding Z2 symmetry Eq. (14), the fields must also comply with them. However A SU 5 → since we are in a gauge theory, the equations need not be invariant and together break 4 completely and ð Þ SUð3Þ × SUð2Þ × Uð1Þ. fulfilled exactly but only up to a gauge transformation, so F; Nc ; ξ any field complies with The fields a;s lie on the brane and are unaffected by the boundary conditions. The fields T are A4 singlets ϕðx; zÞ¼G ϕðx; −zÞ; and do not feel the A4 breaking conditions. They have different parities and feel the SUð5Þ breaking condition. ϕ x; z G5 ϕ x; z 1 ; þ ð Þ¼ ð þ Þ The fields T contain the light MSSM uR, eR fields, while T− Q ϕðx; zÞ¼G6 ϕðx; z þ ωÞ; ð23Þ contains the light field . This allows for independent masses for charged leptons and down quarks since they where the G’s are gauge transformations that must fulfill come from different fields. The Higgs fields feel the SUð5Þ breaking condition leaving only the light doublets, solving G2 1;GG G G ;GGG G−1 ; ¼ 5 6 ¼ 6 5 5;6 ¼ 5;6 ð24Þ the doublet triplet splitting problem [7] (for a recent discussion see e.g., [11]). where the first equation comes from the fact that it belongs The flavons ϕ1;2 feel the A4 breaking conditions. They to the parity operator, the second is due to the fact of the have different parities under the conditions and this fixes commutativity of the translations and the third one denotes their alignments to be the relation between parity and translations. 0 1 0 1 Since the branes z¯i are invariant under the orbifold 1 0 symmetry transformations of Eq. (14), they act as bounda- B C B C hϕ1i¼v1@ 0 A; hϕ2i¼v2@ 1 A: ð29Þ ries which, due to the G’s gauge transformations, impose the boundary conditions 0 1 ϕ x; z z¯ P ϕ x; −z z¯ ; ð þ iÞ¼ z¯i ð þ iÞ ð25Þ We may remark that these flavon vacuum expectation value (VEV) alignments do not break the Z2 symmetry generated which correspond to a reflection at each of the branes. by U, even though they are in the bulk. These boundary conditions are related to the gauge trans- We see that the orbifolding breaks the symmetry formations as SUð5Þ × A4 ⋉ Z2 → SUð3Þ × SUð2Þ × Uð1Þ × Z2 while solving the doublet triplet splitting, separating charged P0 ¼ G; P1=2 ¼ G5G; lepton and down-quark masses and completely aligning P G G; P G G G: flavon VEVs. ω=2 ¼ 6 ð1þωÞ=2 ¼ 5 6 ð26Þ We do not show an explicit driving mechanism for For simplicity, we choose all G’s to commute, meaning that the VEVs v1;2;ξ. We assume that they are driven radia- −1 G5;6 ¼ G5;6, and therefore all boundary conditions become tively [22]. matrices of order 2. The boundary conditions imply an C. Yukawa structure invariance at each brane under some A4 × SUð5Þ transformation, and they are chosen to break the symmetry in a In 6d, the superpotential has dimension 5 while each particular way as follows: superfield has dimension 2. A 6d interacting superpotential

015028-7 DE ANDA, KING, and PERDOMO PHYS. REV. D 101, 015028 (2020) is inherently nonrenormalizable. We work with the effec- are modular forms with a positive even weight [24]. They tive 4d superpotential, which happens after compactifica- involve the Dedekind η function and its exact form can be tion. We assume the compactification scale is close to the found in Appendix D. original cutoff scale. We use Λ to denote both the The modular forms are functions of the lattice basis compactification scale and the GUT scale, which is taken vector parameter τ from Eq. (3). Usually this parameter is to be the cutoff of the effective theory. Assuming this makes chosen to give a good fit to the flavor parameters. In our the Kaluza-Klein modes to be at the GUT scale so that they case, the specific orbifold of our model is set to fix do not spoil standard gauge coupling unification or any of the current precision tests. τ ¼ ω ¼ e2iπ=3; ð32Þ With the fields in Tables I and II, we can write the effective 4d Yukawa terms and the modular form structure is fixed up to a real constant due to the extra condition coming from the generalized CP 3 N c c N ξ c c symmetry. WY ¼ ys ξNsNs þ ya ξ NaNa ν Λ3 The modular form ys must be a triplet under A4 to ξ ϕ ξ construct an invariant singlet with the triplet field F. yν FH Nc yν 2 FH Nc α þ s Λ 5 s þ a Λ2 5 a Furthermore, it has weight to compensate the overall ϕ ϕ ξ ϕ ξ2 weight of the corresponding term. We show the effective ye 1 FH Tþ ye 1 FH Tþ ye 1 FH Tþ triplet alignments it can have in Table III for different þ 3 5¯ 3 þ 2 2 5¯ 2 þ 1 3 5¯ 1 Λ Λ Λ weights α. The possibilities are very limited since many 2 d ϕ1 − d ϕ1ξ − d ϕ1ξ − modular forms vanish when τ ¼ ω, as shown in þ y FH5¯ T þ y FH5¯ T þ y FH5¯ T 3 Λ 3 2 Λ2 2 1 Λ3 1 Appendix D. Larger weight modular forms repeat the same ξ6−i−j structure so that this table is exhaustive, as discussed in yu H TþT− ; þ ij 5 i j 6−i−j ð30Þ Appendix D. Λ ν The modular form ya must have weight β. It multiplies where i, j ¼ 1, 2, 3. Due to the stringent Uð1Þ shaping the flavon ϕ2, so that they must be contracted into ν symmetry, there are no higher order terms. The field ξ has a a triplet ðyahϕ2iÞ3 which will generate the effective align- ν VEV and generates hierarchies between families ala` ment. In the case of ya being a singlet under A4, the Froggat-Nielsen [21]. effective alignment is simply given the flavon VEV hϕ2i in The first line in Eq. (30) gives the two right-handed (RH) Eq. (29), which was fixed by the orbifold boundary ν neutrino Majorana masses without any mixing. The fields conditions. When ya is a triplet under A4, it must be in both terms have zero weight so the modular symmetry contracted with ϕ2 as shown in Appendix B, 3 × 3 → 0 00 does not add anything new. The second line generates Dirac 1 þ 1 þ 1 þ 3a þ 3s. This gives different possible prod- neutrino masses. They have nontrivial weights and their ucts for the effective triplet. The actual effective alignment structure will be discussed in Sec. III D. The third line gives is an arbitrary linear combination of all possibilities and can masses to charged leptons. They are all weight zero be found in Table IV.Forβ ¼ 0 the only modular form is a automatically and the mass matrix is diagonal. The fourth singlet, so the only triplet that can be built is hϕ2i.For ð2Þ line generates a diagonal down-quark mass matrix. Since it β ¼ 2, the only modular form is the triplet Y3 shown in the − involves a different field (T instead of Tþ) the coupling Appendix D. The effective triplet is the linear combination constants are independent. Finally the fifth line gives masses to the up quarks. It is a general nonsymmetric yν mass matrix with complex entries. Since the fields in these TABLE III. The effective alignments of the modular form s as a triplet, depending on its weight α. The parameter y is an T terms have a nontrivial weight but the are singlets, the arbitrary real constant to comply with the extended symmetry modular symmetry does not change the matrix structure. A4 ⋉ Z2. We remark that the top-quark mass term is renormalizable. ν At the GUT level, the μ term is forbidden, so it should be α ðysÞ3 generated by another mechanism at a much smaller 00 ! scale [23]. 2 2 y 2ω −ω2 D. Effective alignments from modular forms ! 4 2 In Eq. (30) we have a few terms involving nontrivial y −ω 2ω2 weights under the modular symmetry. This implies that the ! couplings 6 −1 y 2ω ν ν u 2ω2 ys;ya;yij ð31Þ

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ν TABLE IV. The effective alignments of the modular form ya is easy to check that the transformation in Eq. (33) leaves contracted with hϕ2i into a triplet, depending on its weight β. The the VEV invariant when real and therefore the charged parameters yi are dimensionless constants constrained by the d lepton and down-quark mass terms when the parameters yi A4 ⋉ Z2 symmetry. e and yi involved are real. u ν y α 2γ β ðyahϕ2iÞ3=v2 Finally, the modular form ij must have weight þ to 0 1 build an invariant. All the fields in the corresponding terms 0 0 B C are singlets, so these modular forms must be singlets also y1@ 1 A and will not change the structure. Depending on i, j, the 1 u modular form yij must be a different type of singlet. The 2 ! 2 ! 2 ω − 2ω −ω − 2ω weight α þ 2γ has to be large enough so that the space y 2 y 1 −2ω − 2 þ 2 −2 contains the three types of singlets. This modular form does 4ω − 2 2 ! 2 ! 2 ! not add anything to the structure of the up-quark matrix but 4 1 −2ω þ ω 2ω þ ω A T T 2 allows us to build the 4 invariants for all i j combina- y1ω 0 þ y2 4ω − 2 þ y3 −2 1 −2ω − 2 2 tions. The smallest weight that allows modular forms of all ! ! ! 0 2 2 three types of singlets is 20, as discussed in Appendix D. 6 2ω − 2ω u 2 These modular forms y are in general complex. y1 1 þ y2 4ω þ 1 þ y3 1 ij 1 4ω þ 1 −1 The case β ¼ 0 does not have enough freedom to fit the neutrino data. We conclude that the smallest phenomenologically viable choice for weights is of the symmetric and antisymmetric product of the modular ð2Þ α ¼ β ¼ 6; γ ¼ 7: ð34Þ form with the flavon VEV, hϕ2i × Y3 → 3a þ 3s.For β 4 Yð4Þ;Yð6Þ ¼ , 6 the modular form can be the singlet 10 1 4 6 E. Mass matrix structure respectively and the corresponding triplets Yð Þ;Yð Þ,so 3 3;2 We are now able to express the mass matrices following that the actual alignment comes from the linear combina- Eq. (30) and the effective alignments given in Sec. III D. tion of hϕ2i × Y1;10 → 3 and hϕ2i × Y3 → 3a þ 3s. First, we define the dimensionless parameters By choosing the weights α, β, the structure of the neutrino mass matrix is completely defined. In principle, ˜ hξi=Λ ¼ ξ and vi=Λ ¼ v˜ i; ð35Þ the y in Table III and y1, y2, y3 in Table IV correspond to general complex numbers; however as we will see below where Λ is the original cutoff scale. The down-quark and they are constrained to comply with the nontrivial CP charged lepton mass matrices are diagonal: symmetry of the model. 0 1 We have obtained all the possible A4 invariant modular d ˜2 y1ξ 00 forms. However we have to comply with the extended B C Md v d ˜ v˜ ; symmetry A4 ⋉ Z2. The U generator only transforms ¼ d@ 0 y2ξ 0 A 1 nontrivially the triplet field F which is contracted to a d 00y3 triplet modular form. A U transformation of the field F can 0 1 yeξ˜2 00 be reabsorbed by transforming the modular form by 1 B C Me v e ˜ v˜ ; 0 1 ¼ d@ 0 y2ξ 0 A 1 ð36Þ 100 e B C 00y3 C@ 001A ð33Þ 010 while the up-quark mass matrix can be written as 0 1 u ˜4 u ˜3 u ˜2 where the C stands for complex conjugation. Invariant y11ξ y12ξ y13ξ terms under the full symmetry must involve modular forms B u ˜3 u ˜2 ˜ C Mu ¼ vu@ y21ξ y22ξ y23ξ A; ð37Þ that are also invariant under the Z2 transformation. From u ˜2 u ˜ u Table III, the only invariant case is when α ¼ 6 with a real y31ξ y32ξ y33 y. From Table IV, the only invariant cases happen when yd ye β ¼ 0 with real y1 or β ¼ 6 with y1;2 real and y3 imaginary. where the parameters i and i are real due to the enhanced u The triplet field F is not only taking part in the Dirac symmetry on the branes A4 ⋉ Z2 while yij are in general neutrino mass terms but also in the down-quark and complex. charged lepton mass terms; therefore they also must be The down-quark and charged lepton mass matrices in invariant under the enhanced symmetry A4 ⋉ Z2. In this Eq. (36) are diagonal so the fit to the observed masses is case, the field F is contracted with the flavon field ϕ1 and it straightforward. The hierarchy between the masses of the

015028-9 DE ANDA, KING, and PERDOMO PHYS. REV. D 101, 015028 (2020) different families is understood through the powers of ξ˜ and The neutrino mass matrix looks like can be achieved assuming the dimensionless couplings to v2 ξ˜2 v2 v˜ 2 be of order Oð1Þ. All the contributions to quark mixing m u α α T u 2 β β T; ν ¼ N 6ð 6Þ þ ˜ N 6ð 6Þ ð43Þ come from the up sector. The complex parameters in the hξi ys hξi ξya up-type mass matrix [see Eq. (37)] fix the up, charm and where α6 and β6 are the alignments defined in Eq. (39). The top quark masses as well as the observed Cabibbo- y; y ;y ;y Kobayashi-Maskawa (CKM) mixing angles. We can obtain effective parameters at low energy are f 1 2 3g, previously defined in Tables III and IV. The Z2 symmetry a perfect fit for weight γ ¼ 7. Different values of v˜ 1, v˜ 2 and fixes the parameters fy; y1;y2g to be real while y3 is purely ξ˜ can fit the observed masses using different dimensionless imaginary. couplings still of order O 1 . We show an example in ð Þ Finally we remark that this structure, with the expected Appendix E. hierarchy between the RH neutrinos, can give the correct The form of the Dirac neutrino mass matrix depends on baryon asymmetry of the Universe (BAU) through lepto- the weights α and β. All the possible alignments are given genesis naturally. Leptogenesis is achieved through the CP in Tables III and IV. The Z2 symmetry restricts ourselves to violation in the neutrino Dirac mass matrix. The correct the case α 6 and β 0 or β 6. In the case of β 0,we ¼ ¼ ¼ ¼ order of the BAU happens when the RH neutrino masses only have two free parameters fy; y1g and we cannot find a 10 13 are M1 ∼ 10 GeV and M2 ∼ 10 GeV [26]. In this good fit with the solar and the reactor angle being too small. model, these are the natural expected masses as we can Therefore, the only phenomenologically viable case is for see from Eq. (41) and the sample fit in Appendix E. The α ¼ β ¼ 6 and we restrict ourselves to this case in the contributions from the entries of the neutrino Dirac mass following. matrix and the expected BAU will fix the precise value of As shown in Appendix B, we have to take into account M1. We conclude that the CP violation in the neutrino the Clebsch-Gordan coefficients when contracting the ν ν sector and the RH neutrino mass hierarchy of the model modular form ðysFÞ1 and ðyahϕ2iFÞ1 into singlets, i.e., assures us that the BAU can be generated naturally [11]. 3 × 3 → 1, given by

F. μ − τ reflection symmetry ðφψÞ1 ¼ φ1ψ 1 þ φ2ψ 3 þ φ3ψ 2; ð38Þ The neutrino mass matrix in Eq. (43) is μ − τ reflection after which the effective alignments for α ¼ 6 and β ¼ 6 symmetric (μτ-R symmetric). This corresponds to the look like interchange symmetry between the muon neutrino νμ and 0 1 0 1 the tau neutrino ντ combined with CP symmetry, namely 2 −1 2y2 þ y3ð2ω − 2ωÞ B 2 C B C νe → νe; νμ → ντ ; ντ → νμ; ð44Þ α6 ¼ y@2ω A; β6 ¼ @ y1 þ y2ð4ω þ 1Þ− y3 A; ð39Þ 2 2ω y1 þ y2ð4ω þ 1Þþy3 where the star superscript denotes the charge conjugation of the neutrino field. This can easily be seen from the align- respectively. The Dirac neutrino mass matrix is then ments in Eq. (39) which construct the neutrino mass matrix given by in Eq. (43). Since the parameters fy; y1;y2g are real while y3 0 1 is purely imaginary, the transformation in Eq. (44) leaves the 2 ð2y2 þ y3ð2ω − 2ωÞÞv˜ 2 −y alignments invariant and accordingly the neutrino mass ν B 2 C˜ matrix. For a review of μτ symmetry see e.g., [27] and MD ¼ vu@ ðy1 þ y2ð4ω þ 1Þ − y3Þv˜ 2 2ω y Aξ: ð40Þ references therein; also see the recent discussion [28]. 2 ðy1 þ y2ð4ω þ 1Þþy3Þv˜ 2 2ωy It is known that having a neutrino mass matrix μτ-R symmetric in the flavor basis (which is our case) is equivalent The RH neutrino Majorana mass matrix is diagonal: to μ − τ universal (μτ-U) mixing in the Pontecorvo-Maki- Nakagawa-Sakata (PMNS) matrix; see Ref. [29]. The yNξ˜3 0 μ − τ M ξ a ; consequence of having symmetry is that it leads to R ¼h i N ð41Þ θ 0 ys having a nonzero reactor angle, 13, together with a maximal atmospheric mixing angle and maximal Dirac CP phase: with hierarchical RH neutrino masses given by the different θ ≠ 0; θ 45 ; δl 90 : powers of the field ξ. Furthermore, we have very heavy RH 13 23 ¼ ° ¼ ° ð45Þ neutrino Majorana masses such that the left-handed neu- We remark that this is a prediction of the model, due to trinos get a very small Majorana mass through type I having A4 ⋉ Z2 symmetry on the branes. seesaw [25]: The parameters fy; y1;y2;y3g in the neutrino mass matrix

ν D −1 D T (43) will fit the rest of the PMNS observables, namely mL ¼ Mν MR ðMν Þ : ð42Þ l l 2 2 fθ12; θ13; Δm21; Δm31g, together with the prediction of the

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TABLE V. Two examples with the four input parameters to χ2 ∼ 6800. Therefore, the model predicts normal mass y; y1;y2 and y3 that enter into the neutrino mass matrix in ordering together with maximal atmospheric mixing and Eq. (43), giving the correct PMNS observables. CP violation and a massless neutrino m1 ¼ 0 since we are Fit 1 only adding two RH neutrinos. Parameter Value IV. CONCLUSIONS y −1.28 y1 0.66 In this paper we have presented the first example in the y2 −1.05 literature of a GUTwith a modular symmetry interpreted as a y3 i 1.07 family symmetry. The theory is based on supersymmetric N SU 5 ys 1 ð Þ in 6d, where the two extra dimensions are compacti- N T =Z ya 1 fied on a 2 2 orbifold. We have shown that, if there ˜ jξj 0.1 is a finite modular symmetry, then it can only be A4. jv˜ 2j 0.01 Furthermore, if we restrict ourselves to the case in which jω1j¼jω2j, the only possible value of the modulus param- Fit 2 eter is τ ¼ ω ¼ ei2π=3. We emphasize that this is one of the Parameter Value essential distinctions of the present model in contrast to recent works with modular symmetries, which regard the y −1 00 . modulus τ as a free phenomenological parameter [15,16].In y1 −1.00 the present paper, we assume a specific orbifold structure y2 −0.08 y 0 08 which fixes the modulus to a discrete choice of moduli, 3 i . i2π=3 N where we focus on the case τ ¼ ω ¼ e , although we do ys 1 N not address the problem of moduli stabilization. y 1 a We have shown that it is possible to construct a jξ˜j 0.2 consistent model along these lines, which successfully jv˜ 2j 0.04 combines an SUð5Þ GUT group with the A4 modular symmetry and a Uð1Þ shaping symmetry. In this model the l μ − τ symmetry, θ23 ¼ 45° and δ ¼ −90°. The contribution F fields on the branes are assumed to respect an enhanced 2 to a χ test function comes only from these predictions and symmetry A4 ⋉ Z2 which leads to an effective μ − τ we use the recent global fit values of neutrino data from reflection symmetry at low energies, which predicts the NuFit4.0 [30]. The best fit points together with the 1σ ranges maximal atmospheric angle and maximal CP phase. In θ =∘ 49 6þ1.0 δl=∘ 215þ40 addition there are two right-handed neutrinos on the branes are 23 ¼ . −1.2 and ¼ −29 for normal mass ordering and without the Super-Kamiokande atmospheric whose Yukawa couplings are determined by modular neutrino data analysis. However, the distributions of these weights, leading to specific alignments which fix the two observables are far from Gaussian and the predictions of Dirac mass matrix. The model also introduces two triplet having maximal atmospheric mixing angle θ23 ¼ 45° and flavons in the bulk, whose vacuum alignments are deter- maximal CP violation δl ¼ −90° still lie inside the 3σð4σÞ mined by orbifold boundary conditions, analogous to those region with a χ2 ¼ 5.48 (6.81) without (with) Super- responsible for Higgs doublet-triplet splitting. The charged Kamiokande. Appendix E explains how a numerical fit lepton and down-type quarks have diagonal and hierarchi- can be performed and Table V shows two numerical fits, cal Yukawa matrices, with quark mixing due to a hierar- although this is only an example as we can find a good fit for chical up-quark Yukawa matrix. 4 a large range of parameters y; y1;y2 and y3. This is because The resulting model, summarized in Tables I and II, l provides an economical and successful description of quark the predictions of the model θ23 ¼ 45° and δ ¼ −90° are due to the μτ-R symmetry and the four free parameters are and lepton (including neutrino) masses and mixing angles CP used to fit the rest of the observables in the PMNS matrix. and phases. Indeed the quarks can be fit perfectly, SU 5 O 1 The best fit from NuFit4.0 is for normal mass consistently with ð Þ, using only ð Þ parameters. In ordering with inverted ordering being disfavored with a addition we obtain a very good fit for the lepton observ- χ2 ≈ 5 7 Δχ2 ¼ 4.7 ð9.3Þ without (with) the Super-Kamiokande ables with ð Þ without (with) Super-Kamiokande O 1 atmospheric neutrino data analysis. We tried a fit to data, using four ð Þ parameters which determine the U inverted mass ordering and the χ2 test function goes up entire lepton mixing matrix PMNS and the light neutrino masses (eight observables), which implies that the theory is quite predictive. The main predictions of the model are a 4 α 0 β 0 Although the model only allows the weights ¼ and ¼ , normal neutrino mass hierarchy with a massless neutrino, 6, we tried a numerical fit with all possible combinations of μ − τ θl 45 weights with the alignments in Tables III and IV, and the only one and the reflection symmetry predictions 23 ¼ ° l that worked is the μτ-R symmetric for α ¼ β ¼ 6. and CP phase δ ¼ −90°, which will be tested soon.

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ACKNOWLEDGMENTS where p, q are general integer numbers and represent the general orbifold transformations. This relates the basis We thank Patrick Vaudrevange for useful discussions. vectors ω1 and ω2 as S. F. K. acknowledges the STFC Consolidated ’ Grant No. ST/L000296/1 and the European Union s e−i2π=M − 2p − 1 Horizon 2020 Research and InvisiblesPlus RISE Grant ω ω ; 2 ¼ 2q 1 1 ðA5Þ No. 690575. E. P. acknowledges the European Union’s þ Horizon 2020 Research. S. F. K. and E. P. acknowledge the and we may rewrite the transformed branes from Eq. (A3) Innovation program under Marie Skłodowska-Curie grant in terms of ω1: agreement Elusives ITN Grant No. 674896. e−i2π=M 1 1 − 2p 1 e2iπ=M 0 ð þ Þ z¯T ¼ 0; ; þ ; 2 2 4q þ 2 APPENDIX A: UNIQUENESS OF A4 AS A e−i2π=M 1 1 − 2p 1 e2iπ=M MODULAR SYMMETRY FOR THE BRANES ð þ Þ ω : 2 þ 2 þ 4q 2 1 ðA6Þ In this Appendix, we show that the set of branes in þ ¯ Eq. (11) is invariant under the modular transformations Γ3 We now make the Ansatz that the third brane corresponds to for an infinite set of discrete values of the modulus the third original brane ω2=2 (and automatically the fourth parameter τ. corresponds to the original second one), so that it must First, we apply the finite modular transformations in satisfy Eq. (9) on the lattice vectors 1 − ð2p þ 1Þe2iπ=M e−i2π=M − 2p − 1 ω1 ω2 1 þ ¼ 2r þð2s þ 1Þ ; S ¼ ; 2q þ 1 2q þ 1 ω2 −ω1 ðA7Þ ω e−2iπ=Mω T 1 1 : ðMÞ ¼ 2iπ=M ðA1Þ r s ω2 ω1 þ e ω2 where , are general integer numbers which represent the general orbifold transformations. After some simple manip- Therefore, the S-transformed branes become ulations this can be written as 2s 1 e−2iπ=M 2p 1 e2iπ=M ω −ω ω − ω ð þ Þ þð þ Þ z¯0 0; 2 ; 1 ; 2 1 : S ¼ 2 2 2 ðA2Þ ¼ð1 − 2rÞð2q þ 1Þþð2p þ 1Þð2s þ 1Þþ1: ðA8Þ

Using the orbifold transformations from Eq. (10), we can The left-hand part is complex while the right-hand side is real. To cancel the imaginary part we must have add ω1 to the second and fourth branes, and obtain the original set in Eq. (11). Therefore the brane set is always s ¼ p; ðA9Þ invariant under the S transformation, for any value of ω1 and ω2. On the other hand, the T-transformed branes are so that 2 2p 1 cos 2π=M e−2iπ=Mω ω e2iπ=Mω ð þ Þ ð Þ z¯0 0; 1 ; 1 þ 2 ; T ¼ 2 2 ¼ð1 − 2rÞð2q þ 1Þþð2p þ 1Þð2p þ 1Þþ1; ðA10Þ −2iπ=M 2iπ=M e ω1 þ ω1 þ e ω2 : ðA3Þ where, since the right-hand side is an integer, the left-hand 2 side must also be which forces

If the set is to be invariant, up to permutations of the M ¼ 3; 6; ðA11Þ branes, the second term in Eq. (A3) must correspond to one of the original branes. and the equation becomes Let us make the Ansatz that the second brane from Eq. (A3) corresponds to the fourth original brane mð2p þ 1Þ¼ð1 − 2rÞð2q þ 1Þþð2p þ 1Þð2p þ 1Þþ1; ω ω =2 ð 1 þ 2Þ up to orbifold transformations, so that it must where m ¼ð−1ÞM; ðA12Þ satisfy which is a Diophantine equation to be solved for r.Wehave e−i2π=Mω 2p 1 ω 2q 1 ω ; 1 ¼ð þ Þ 1 þð þ Þ 2 ðA4Þ made two Ansätze to obtain this equation. This is the only

015028-12 SUð5Þ GRAND UNIFIED THEORY WITH A4 MODULAR … PHYS. REV. D 101, 015028 (2020) solution since making any other Ansatz would obtain an TABLE VII. Decomposition of the product of two triplets φ; ψ. equation without solutions (the equation would be an odd number equal to zero). These are straightforward calcu- Component decomposition lations, which are done in the same way as this one and ðφψÞ1 φ1ψ 1 þ φ2ψ 3 þ φ3ψ 2 φψ φ ψ φ ψ φ ψ seem repetitive to show. ð Þ10 3 3 þ 1 2 þ 2 1 φψ φ ψ φ ψ φ ψ The invariance condition on the branes fixes them to be: ð Þ100 2 2 þ 3 1 þ 1 3 ! ðφψÞ3 2φ1ψ 1 − φ2ψ 3 − φ3ψ 2 s 1 i2π=3 pﬃﬃ 2φ ψ − φ ψ − φ ψ −e − 2p − 1 m − 1 =2 3 3 3 1 2 2 1 ω þð Þ ω ; 2φ ψ − φ ψ − φ ψ 2 ¼ 1 ðA13Þ 2 2 3 1 !1 3 2q þ 1 φψ φ ψ − φ ψ ð Þ3a 2 3 3 2 φ1ψ 2 − φ2ψ 1 where the choice M ¼ 3, 6 only changes with m. Since φ3ψ 1 − φ1ψ 3 only ω2 is physical (and not the specific integers p, q), we can reabsorb the m dependence into p. As we stated before we will be studying discrete modular symmetries with M ≤ 5 so that the only solution is M ¼ 3 which fixes The component decompositions of the products are shown the relation of the branes to be in Table VII. The 12 elements of A4 are obtained as 1;S;T;ST; i2π=3 2 2 2 2 2 e þ 2p þ 2 TS;T ;ST ;STS;TST;T S; TST and T ST. The A4 ele- ω2 ¼ − ω1; ðA14Þ 2q þ 1 ments belong to four conjugacy classes: where the p, q are integers that satisfy that 1C1∶1 4C ∶T; ST; TS; STS ð2p þ 1Þðp þ 1Þþq þ 1 3 r ¼ ðA15Þ 4C2∶T2;ST2;T2S; ST2S 2q þ 1 3 2 2 3C2∶S; T ST; TST ; ðB2Þ is an integer, which has infinitely many discrete solutions. k where mCn refers to the Schoenflies notation where m is APPENDIX B: GROUP THEORY the number of elements of rotations by an angle 2πk=n.

A4 is the even permutation group of four objects, which is isomorphic to the symmetry group of a regular tetrahe- APPENDIX C: GENERALIZED CP dron. It has 12 elements that can be generated by two CONSISTENCY CONDITIONS FOR A generators, S and T, with the presentation 4 Here, we check the compatibility of the Z2 symmetry on 2 3 3 S ¼ T ¼ðSTÞ ¼ 1: ðB1Þ the branes with the A4 flavor symmetry. The remnant Z2 symmetry behaves as an effective generalized CP trans- A4 has four inequivalent irreducible representations: three formation and the fields on the branes will transform under 1; 10; 100 3 singlet and one triplet representations. We choose Z2 as to work with the same complex basis as [15] and the 0 representation matrices of the generators are shown in ψðxÞ → Xrψ ðx Þ; ðC1Þ Table VI. φ φ ; φ ; φ ψ 0 The product of two triplets, ¼ð 1 2 3Þ and ¼ where x ¼ðt; x1;x2;x3;x5; −x6Þ and Xr is the representa- 0 00 ðψ 1; ψ 2; ψ 3Þ, decomposes as 3×3¼1þ1 þ1 þ3s þ3a, tion matrix in the irreducible representation r. To combine 3 where s;a denote the symmetric or antisymmetric product. the flavor symmetry A4 with the Z2 symmetry, the transformations have to satisfy certain consistency conditions A TABLE VI. Generators S and T in the irreducible representa- [31], which were specifically applied to 4 flavor sym- 2πi=3 tions of A4, where ω ¼ e . metry in [20]. These conditions assure that if we perform a Z2 transformation, then apply a family symmetry trans- A 110 100 3 4 formation, and finally an inverse Z2 transformation is ! S 11 1 −12 2 followed, the resulting net transformation should be equiv- 1 3 2 −12 alent to a family symmetry transformation. It is sufficient to 22−1 ! only impose the consistency conditions on the group T 1 ω ω2 10 0 generators: 0 ω 0 2 00ω −1 0 −1 0 XrρrðSÞXr ¼ ρrðS Þ;Xrρr ðTÞXr ¼ ρrðT Þ; ðC2Þ

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2 where ρr denotes the representation matrix for the gen- Y → ðcτ þ dÞ ρY: ðD3Þ erators S and T; see Table VI. As shown in [20], S0 and T0 can only belong to certain conjugacy classes of A4, The only modular forms of weight 2 behave as a triplet Y 0 0 2 and can be written as S ∈ 3C2;T∈ 4C3 ∪ 4C3; ðC3Þ [see Eq. (B2) to find out the elements in each conjugacy 0 τ 0 τþ1 0 τþ2 0 i η ð3Þ η ð 3 Þ η ð 3 Þ 27η ð3τÞ class]. The transformations under the generalized CP Y1ðτÞ¼ þ þ − ; 2π η τ η τþ1 η τþ2 η 3τ Z ð3Þ ð 3 Þ ð 3 Þ ð Þ symmetry 2 are then η0 τ η0 τþ1 η0 τþ2 0 1 −i ð3Þ 2 ð 3 Þ ð 3 Þ 100 Y2ðτÞ¼ þ ω þ ω ; π η τ η τþ1 η τþ2 B C ð3Þ ð 3 Þ ð 3 Þ ψ 10 → ψ 00 ; ψ 100 → ψ 0 ; ψ 3 → @001Aψ ; C4 1 1 3 ð Þ η0 τ η0 τþ1 η0 τþ2 −i ð3Þ ð 3 Þ 2 ð 3 Þ 010 Y3ðτÞ¼ þ ω þ ω ; ðD4Þ π η τ τþ1 τþ2 ð3Þ ηð 3 Þ ηð 3 Þ which are consistent with Eqs. (C2) and (C3) for S0 ¼ S and T0 T ¼ . However in the model under consideration, we do where the ηðτÞ denotes the Dedekind function not have any field on the branes transforming under the 10 00 and 1 representation. Thus the Z2 transformation only Y∞ affects the 3 representations. η τ q1=24 1 − qn ;q≡ ei2πτ: D5 We conclude that the 3 representations on the brane ð Þ¼ ð Þ ð Þ n¼1 transform under A4 ⋉ Z2 as shown in Table VI and Eq. (C4). There are no weight 2 singlets. In our model, the modulus field is fixed by the orbifold to be τ ¼ ω. In this case, up to APPENDIX D: MODULAR FORMS an overall coefficient, we have In this section, we show the construction of modular Γ¯ ≃ A 2 forms for 3 4 following [15]. Y1ðωÞ¼2;Y2ðωÞ¼2ω;Y3ðωÞ¼−ω ; ðD6Þ In model building, the difference between the usual cited discrete symmetries, which arise as remnant symmetries of the branes, and the modular symmetries of the spacetime and the triplet for weight 2 is lattice is that, in the latter case, the fields transform, under a τ transformation of ,as ð2Þ 2 Y3 ¼ð2; 2ω; −ω Þ: ðD7Þ ψ → ðcτ þ dÞ−kρψ; ðD1Þ Higher weight modular forms can be written in terms of the ρ where is the usual matrix representation of the trans- weight 2 forms by taking products of them. The weight 4 k formation and is called the weight and is an arbitrary modular forms are written as number. The invariance of the action forces the usual dimensionless coupling in the superpotential y to behave ð4Þ 2 2 2 as [32] Y3 ¼ðY1 − Y2Y3;Y3 − Y1Y2;Y2 − Y1Y3Þ; ð4Þ 2 ky Y Y 2Y Y ; y → ðcτ þ dÞ ρyy; ðD2Þ 1 ¼ 1 þ 2 3 Yð4Þ Y2 2Y Y ; 10 ¼ 3 þ 1 2 where ky is the weight and must be an even integer [24] and ρ Yð4Þ Y2 2Y Y ; y the usual matrix representation of the transformation. To 100 ¼ 2 þ 1 3 ðD8Þ build the invariant in global supersymmetry, we need to satisfy two conditions: first the weight ky has to cancel the where the subscript corresponds to the representation under overall weights of the fields and second the product of ρy A4. In our model, the modulus field is fixed by the orbifold times the representation matrices of the fields has to contain to be τ ¼ ω. In this case, the only nonzero weight 4 an invariant singlet. When k 0 for every constant, we ¼ modular forms are have the usual discrete symmetry. The weight 0 form is just a constant, singlet under A4. The first nontrivial modular form is of weight 2 that Yð4Þ 2; −ω; 2ω2 ;Yð4Þ ω: 3 jτ ω ¼ð Þ 10 ¼ ðD9Þ following Eq. (4) transforms as ¼

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The weight 6 modular forms are written as APPENDIX E: NUMERICAL FIT We perform a χ2 test function when fitting the effective ð6Þ 3 3 3 neutrino mass matrix in Eq. (43) with input parameters Y1 ¼ Y1 þ Y2 þ Y3 − 3Y1Y2Y3; x ¼ y; y1;y2;y3, from which we obtain a set of observables ð6Þ 3 2 2 2 2 Y3;1 ¼ðY1 þ 2Y1Y2Y3;Y1Y2 þ 2Y2Y3;Y1Y3 þ 2Y3Y2Þ; PnðxÞ. We minimize the function defined as ð6Þ 3 2 2 2 2 Y Y 2Y1Y2Y3;Y Y1 2Y Y2;Y Y2 2Y Y1 ; 3;2 ¼ð 3 þ 3 þ 1 3 þ 2 Þ X obs 2 2 PnðxÞ − Pn ð6Þ 3 2 2 2 2 χ ¼ ; ðE1Þ Y Y 2Y1Y2Y3;Y Y3 2Y Y1;Y Y1 2Y Y3 : σ 3;3 ¼ð 2 þ 2 þ 3 2 þ 1 Þ n n ðD10Þ obs l l l l where the observables are given by Pn ∈ fθ12; θ13; θ23; δ ; Δm2 ; Δm2 σ Due to relations of the Dedekind functions, the modular 21 31g with statistical errors n. We use the recent forms satisfy global fit values of neutrino data from NuFit4.0 [30] and we ignore any renormalization group running corrections as well as threshold corrections associated with the two extra Y2 2Y Y 0; 2 þ 1 3 ¼ ðD11Þ dimensions. Most of the observables follow an almost Gaussian distribution and we take a conservative approach which reduce the number of possible modular forms. In our using the smaller of the given uncertainties in our compu- l l case tations except for θ23 and δ . The best fit from NuFit4.0 is for normal mass ordering with inverted ordering being 2 2 disfavored with a Δχ ¼ 4.7 ð9.3Þ without (with) the ðY þ 2Y2Y3Þjτ ω ¼ 0; ðD12Þ 1 ¼ Super-Kamiokande atmospheric neutrino data analysis. We tried a fit to inverted mass ordering and we found a which reduces even further the possible modular forms. χ2 ∼ 6800; therefore in the following results we only focus The only triplet that is different from zero in Eq. (D10) is on the case of normal mass ordering. The model predictions are shown in Table VIII. The ð6Þ 2 Y jτ ω ¼ð−1; 2ω; 2ω Þ: ðD13Þ neutrino mass matrix in Eq. (43) predicts the maximal 3;2 ¼ l atmospheric mixing angle, θ23 ¼ 45°, and maximal CP δl −90 3σ All modular forms are built from products of the weight violation, ¼ °, within the region from the latest μτ 2 triplet. We can build the modular forms for weight 8. neutrino oscillation data. This is a consequence of the -R y; y ;y Following [15], this is a 15-dimensional space that must be symmetric form of the neutrino mass matrix when 1 2 decomposed as 2 × 1 þ 2 × 10 þ 2 × 100 þ 3 × 3. For simplicity we can work out only the specific case where τ ¼ ω. TABLE VIII. Model predictions in the neutrino sector for This case is greatly restricted and can be checked by doing weights α ¼ β ¼ 6. The neutrino masses mi as well as the all possible multiplications of 3 × 3 × 3 × 3 that the only Majorana phases are pure predictions of our model. We also θl 45 nonzero modular forms are predict the maximal atmospheric mixing angleP 23 ¼ ° and l maximal CP phase δ ¼ 270°. The bound on mi is taken from [34]. The bound on mee is taken from [35]. There is only one Yð8Þ 2; 2ω; −ω2 ;Yð8Þ ω2; 3 ¼ð Þ 100 ¼ ðD14Þ physical Majorana phase since m1 ¼ 0.

Data where we can see that the triplet has the same structure as Model 1σ α β 6 the weight 2 one. From this we conclude that any higher Observable Central value range ¼ ¼ l ° 33 06 → 34 60 weight triplet would only repeat the previous structures θ12 ð Þ 33.82 . . 33.82 l ° 8 480 → 8 740 without having any new one. θ13 ð Þ 8.610 . . 8.610 l ° 48 40 → 50 60 For weight 10 we would have the same triplet as in θ23 ð Þ 49.60 . . 45.03 weight 4 but two singlets since we can have the nontrivial δl ð°Þ 215.0 186.0 → 255.0 270.00 2 −5 2 7 190 → 7 600 products Δm21=ð10 eV Þ 7.390 . . 7.390 2 −3 2 Δm31=ð10 eV Þ 2.525 2.493 → 2.558 2.525 m1 (meV) 0 Yð6Þ Yð4Þ → 10;Yð6Þ Yð4Þ → 100; 1 × 10 3 × 3 ðD15Þ m2 (meV) 8.597 Pm3 (meV) 50.25 m ≲230 so that this is the first space that has two singlets. The next i (meV) 58.85 α ° 180.00 space that has the three singlets is built from powers of 23 ð Þ mee (meV) ≲60–200 2.587 these singlets, so the modular form must have weight 20.

015028-15 DE ANDA, KING, and PERDOMO PHYS. REV. D 101, 015028 (2020) are real while y3 is imaginary. Furthermore, since we only TABLE IX. Input parameters that enter in the charged fermion have 2RH neutrinos, m1 ¼ 0 and there is only one physical mass matrices in Eqs. (36) and (37), giving the correct charged ξ˜ Majorana phase α23 [33]. The predicted effective Majorana fermion masses and CKM parameters for a choice of correspond- yu mass mee [33] is also given in Table VIII. ing to fit 1. The Yukawa parameters ij are in general complex; The fit has been performed using the Mixing Parameter however most of the phases can be reabsorbed and we are left with only four physical phases ϕ21, ϕ23, ϕ31 and ϕ32 where the subscript Tools (MPT) package [36]. The values of y; y1;y2 and y3 are shown in Table V. Fit 1 shows a good fit where all of the refers to the entry in the mass matrix wherewe are adding the phase. y O 1 dimensionless real parameters are of ð Þ. However a Fit 1 large range of parameters can give an equally good fit; see ˜ Parameter Value for example fit 2. The VEV ratios jξ; v˜ ij are parameters that yu do not enter the fit directly and they are chosen to reproduce 11 1.09 yu the hierarchy between the fermion Yukawa couplings, 12 0.62 yu −0 44 making the dimensionless couplings more natural numbers, 13 . yu −0 42 O 1 21 . i.e., ð Þ. In the case of the neutrino mass matrix, even for u ˜ y22 −0.23 fixed jξj and jv˜ 2j, there is a large range of parameters u y23 0.21 y; y1;y2 and y3 that can give a good fit to the observables, u y31 −1.24 meaning that the modular forms for weight α ¼ 6 and β ¼ u y32 −0.56 6 u give a constrained form of the neutrino mass matrix y33 0.54 which is phenomenologically suitable. For comparison, we ϕ =° 354.47 2 21 χ ° also give the value of the test function in the case of ϕ23= 354.67 β 0 y y ° ¼ , in which we only have two free parameters and 1, ϕ31= 328.83 2 χ ∼ 1500 β 6 ° and it goes up to , while for ¼ with four free ϕ32= 329.49 parameters we have found a perfect fit for a variety of jξ˜j 0.1 values of y; y1;y2 and y3. ˜ ˜ jv1j 0.01 These VEV ratios jξ; v˜ ij also appear in the quark and charged-lepton mass matrices in Eqs. (36) and (37).For Fit 1 different values of jξ˜j, as in fits 1 and 2 in Table V, different d e u Parameter Value dimensionless Oð1Þ parameters yi ;yi and yij can be used yd to give the correct mass of the down- and up-type quarks 1 0.25 yd and charged leptons and we show an example in Table IX 2 0.49 yd for fit 1. In this case we take into account the running of the 3 2.74 ye MSSM Yukawa couplings to the GUT scale and we follow 1 0.10 ye the parametrization done by [37]. The matching conditions 2 2.12 ye at the SUSY scale are parametrized in terms of four 3 3.60 ˜ parameters η¯q;b;l which we set to zero and the usual jξj 0.1 tan β for which we choose tan β ¼ 5. jv˜ 1j 0.01

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