CP Symmetry and Symplectic Modular Invariance
UUSTC-ICTS/PCFT-21-07
CP Symmetry andSymplectic Modular Invariance
Gui-Jun Ding , ∗ , Ferruccio Feruglio § and Xiang-Gan Liu , ‡ Peng Huanwu Center for Fundamental Theory, Hefei, Anhui 230026, China Interdisciplinary Center for Theoretical Study and Department of Modern Physics,University of Science and Technology of China, Hefei, Anhui 230026, China Dipartimento di Fisica e Astronomia ‘G. Galilei’, Universit`a di PadovaINFN, Sezione di Padova, Via Marzolo 8, I-35131 Padua, Italy
Abstract
We analyze CP symmetry in symplectic modular-invariant supersymmetric the-ories. We show that for genus g ≥ g ≤
2. We discuss the transformationproperties of moduli, matter multiplets and modular forms in the Siegel upperhalf plane, as well as in invariant subspaces. We identify CP-conserving surfacesin the fundamental domain of moduli space. We make use of all these elements tobuild a CP and symplectic invariant model of lepton masses and mixing angles,where known data are well reproduced and observable phases are predicted interms of a minimum number of parameters. ∗ E-mail: [email protected] § E-mail: [email protected] ‡ E-mail: [email protected] a r X i v : . [ h e p - ph ] F e b Introduction
Fermion masses, mixing angles and CP violating phases are tightly linked together in thepresent picture of particle interactions. Yet a fundamental principle explaining their ori-gin and allowing a more economical and basic description is still lacking. The most widelyexplored approach to the problem is based on flavour symmetries invoked to constrain La-grangian parameters [1]. No exact flavour symmetry does the job, however, and realisticmodels should rely heavily on the properties of the symmetry breaking (SB) sector, com-prising a set of scalar fields whose vacuum expectation values (VEV) are suitably orientedin flavour space. To reduce the vast arbitrariness associated with this construction, whereboth the flavour group and the SB sector are essentially unrestricted, we recently proposed aframework defined by a set of geometrical data [2]. Scalars responsible for SB span a modulispace, a symmetric space of the type
G/K , G being a noncompact continuous group and K a maximal compact subgroup of G . A discrete, modular subgroup Γ of G , acting on G/K ,plays the role of flavour symmetry. In this way both the flavour symmetry group and theSB sector are closely linked and cannot be arbitrarily chosen. Modular invariant supersym-metric field theories [3, 4] are a particular case of this general setting, related to the choice G = SL (2 , R ), K = SO (2) and Γ = SL (2 , Z ). The moduli space G/K is the upper half planeand Yukawa couplings are classical modular forms [5].This bottom-up proposal is evidently inspired by top-down considerations from stringtheory. In string theory Yukawa couplings are indeed field dependent quantities, specified bythe background over which the string propagates. A substantial part of this background isof geometrical origin and is described by moduli, scalar fields belonging to the moduli space,which is often a symmetric space of the type
G/K [6]. A wealth of theoretical activity has infact its focus on the study of Yukawa couplings in realistic string theory compactifications [7–17] and their modular properties [18–38]. Moreover, in string theory finite modular invarianceis in general only a component of a bigger
Eclectic Flavour Group , which also involves CPand an ordinary flavour group leaving moduli invariant [39–44].Inclusion of CP transformations is an important step in a comprehensive description ofparticle properties. If the theory is CP-invariant, CP violation can arise only as a conse-quence of the choice of the vacuum. Then, in the previously discussed class of theories, CPproperties depend on the chosen point in moduli space, which simultaneously controls parti-cle masses and mixing angles. In this context most of the observed features of the fermionspectrum might be determined mostly by the vacuum, rather than by Lagrangian parameters.Moreover, the origin of fermion masses, mixing angles and phases can be fully unified.In the presence of a discrete symmetry like the modular one, a consistent definition ofCP is not granted and requires the existence of nontrivial automorphisms of the symmetrygroup [45–47]. Consistent CP transformations in modular invariant supersymmetric theo-ries dealing with a single modulus have been discussed in refs. [39, 40]. In particular, CPtransformation laws of the modulus τ , of chiral matter multiplets and of modular forms havebeen determined [48] and several models where CP is spontaneously broken have been con-structed [48–54]. Within the more general case of a multidimensional moduli space, consistentCP definitions have been examined recently in the context of symplectic modular invarianttheories, where the relevant flavour group is the Siegel modular group [55, 56]. Ref. [55]2iscusses also CP-conserving vacua in Calabi-Yau compactifications.In this work, we reconsider CP invariance in symplectic modular invariant theories, froma bottom-up perspective. We examine thoroughly all candidate CP definitions, arising asnon-trivial automorphisms of the Siegel modular group Γ = Sp (2 g, Z ), which coincides with SL (2 , Z ) when g = 1. We show that for genus g ≥ g ≤ . Moreover, beyond the action of CP onmoduli space, we analyze also the correct CP transformations properties of matter multipletsand of Siegel modular forms. We also show that, beyond the surface Re ( τ ) = 0, there areinfinite CP-invariant points on the boundary of the Siegel fundamental domain. These arethe ingredients needed to build concrete multi-moduli CP-invariant models and, in the finalpart of our paper, we propose one such model describing the lepton sector at genus g = 2.By making use of a minimum number of Lagrangian parameters (five, to describe twelveobservable quantities), our model reproduces all known data and predicts the CP violatinglepton phases, with moduli intriguingly close to a point of enhanced symmetry in modulispace.Our paper consists of seven Sections. In Section 2, we start by reviewing the formalism ofsymplectic modular-invariant supersymmetric theories. In Section 3, we provide an extensivediscussion of the possible CP definitions in such theories. In Section 4 we study CP-conservingpoints in moduli space and in Section 5 we formulate the correct definition of CP when thetheory is restricted to an invariant subspace of the entire moduli space. Our model is builtand analyzed in Section 6. In a final Section we present our conclusion. In the class of theories under consideration here, both the flavour symmetry and the fieldsresponsible for SB have the same origin [2]. Scalars driving SB take values in a symmetricspace of the type
G/K , G being some noncompact continuous group and K a maximalcompact subgroup of G . The flavour symmetry group is a discrete, modular subgroup Γof G , acting on G/K . Symplectic modular invariant supersymmetric theories are based onthe choice G = Sp (2 g, R ), K = Sp (2 g, R ) ∩ O (2 g, R ) = U ( g ) and Γ = Sp (2 g, Z ). Theinteger g ( g = 1 , , ... ) is called genus. The moduli space H g = Sp (2 g, R ) /U ( g ) is the Siegelupper half plane, a natural generalization of the well-known complex upper half plane H .The Siegel modular group Γ = Sp (2 g, Z ) arises as the duality group in string Calabi-Yaucompactifications [57–64]. Siegel modular forms are relevant in the context of string one-loopcorrections [65, 66].The elements of the symplectic group Sp (2 g, R ) are 2 g × g real matrices of the type γ = A BC D , (2.1) For g = 1, this alternative had already been emphasized in ref. [50]. γ t J γ = J , where J is the symplectic form J : J = g − g . (2.2)For g = 1, the symplectic condition on a matrix is satisfied if and only if the determinant isone, so that we have Sp (2 , R ) = SL (2 , R ). An element τ of the moduli space H g is describedby a symmetric complex g × g matrix τ with positive definite imaginary part: H g = (cid:110) τ ∈ GL ( g, C ) (cid:12)(cid:12)(cid:12) τ t = τ, Im ( τ ) > (cid:111) . (2.3)Similarly to the case G = SL (2 , R ), the action of Sp (2 g, R ) on τ is defined as: τ → γτ = ( Aτ + B )( Cτ + D ) − . (2.4)As modular group Γ we can choose a discrete subgroup of Sp (2 g, R ). A reference choice is theSiegel modular group Γ g = Sp (2 g, Z ), obtained from Sp (2 g, R ) by restricting the elementsof the matrices A , B , C and D in eq. (2.1) to integer values. A set of generators for Γ g is { S, T i } : S = g − g , T i = g B i g , (2.5)where { B i } is a basis for the g × g integer symmetric matrices and S coincides with theinvariant symplectic form J satisfying S = − g . In particular, for g = 1, the Siegelmodular group Sp (2 , Z ) coincides with the special linear group SL (2 , Z ) and the generatorsof Γ are S = − , T = . (2.6)For the case of g = 2, it is convenient to choose the generators of Γ as S = − , T = B , T = B , T = B , (2.7)with B = , B = , B = . (2.8)Under S and T i transformations, we find: τ S −→ − τ − , τ T i −→ τ + B i . (2.9)4ther discrete subgroups of G = Sp (2 g, R ), of direct interest here, are the principal congru-ence subgroups Γ g ( n ) of level n , defined as:Γ g ( n ) = (cid:110) γ ∈ Γ g (cid:12)(cid:12)(cid:12) γ ≡ g mod n (cid:111) , (2.10)where n is a generic positive integer, and Γ g (1) = Γ g . The group Γ g ( n ) is a normal subgroupof Γ g , and the quotient group Γ g,n = Γ g / Γ g ( n ), which is known as finite Siegel modular group,has finite order [67, 68]. Symplectic modular invariance can be seen as a gauge symmetry related to the redundancyof the description of physical vacua. Thus it is useful to introduce a fundamental domain F g describing the set of inequivalent vacua. This is essentially the quotient between H g and Γ g .More precisely, a fundamental domain F g in H g for the Siegel modular group Γ g is a connectedregion of H g such that each point of H g can be mapped into F g by a Γ g transformation, butno two points in the interior of F g are related under Γ g . It is considerably more complicatedthan the g = 1 case. A fundamental domain F g for the action of Γ g on H g can be defined asfollows [69]: F g = τ ∈ H g (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Im ( τ ) is reduced in the sense of Minkowski , | det( Cτ + D ) | ≥ γ ∈ Γ g , | Re ( τ ij ) | ≤ / . (2.11)Here Minkowski reduced means that Im ( τ ) satisfies the two properties:1) h t Im ( τ ) h ≥ Im ( τ ) kk ( ∀ h = ( h , . . . , h g ) ∈ Z g ) for 1 ≤ k ≤ g whenever h , . . . , h g arecoprime ;2) Im ( τ ) k,k +1 ≥ ≤ k ≤ g − g such a fundamental domain is determinedby only finitely many inequalities of the form | det( Cτ + D ) | ≥ ∂ F g is defined as the set of points in F g , where at least one of therelations in eq. (2.11) is realized as an equality. In general points lying on the boundary ∂ F g are related by Siegel modular transformations. The boundary of the fundamental domain F g for g > g = 2 we parametrize the moduli τ as τ = τ τ τ τ . (2.12)5he fundamental domain F can be defined by the following inequalities [70, 71]: F = τ ∈ H (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) | Re ( τ ) | ≤ / , | Re ( τ ) | ≤ / . | Re ( τ ) | ≤ / , Im ( τ ) ≥ Im ( τ ) ≥ Im ( τ ) ≥ , | τ | ≥ , | τ | ≥ , | τ + τ − τ ± | ≥ , | det( τ + E i ) | ≥ , , (2.13)where the set {E i } includes the following 15 matrices: , ± , ± , ± ± , ± ∓ , ± ± , ± ± ± , ± ± ± . (2.14)When one of these inequality is satisfied as an equality, we recover a polynomial equation in Re ( τ i ) and Im ( τ i ), defining a real 5-dimensional wall. From eq. (2.13) we count 28 walls, thatdetermine the boundary ∂ F of the domain F . Another important building block of the theory are the Siegel modular forms, holomorphiccomplex functions of the variables τ , enjoying good transformation properties under theSiegel modular group Γ g . They are specified by the genus g , the weight k and the level n , k and n being non-negative integers. Siegel modular forms Y ( τ ) of integral weight k and level n at genus g are holomorphic functions on the Siegel upper half plane H g transforming underΓ g ( n ) as Y ( γτ ) = [det( Cτ + D )] k Y ( τ ) , γ = A BC D ∈ Γ g ( n ) . (2.15)When n = 1 ,
2, we have − g ∈ Γ g ( n ) and the above definition gives [2]: Y ( − g τ ) = Y ( τ ) = ( − kg Y ( τ ) . (2.16)Therefore Siegel modular forms at genus g of weight k and level n = 1 , kg is odd.The complex linear space M k (Γ g ( n )) of Siegel modular forms of given weight k , level n andgenus g is finite dimensional and there are no non-vanishing forms of negative weight [72].Similarly to the case g = 1 [5], it is possible to choose a basis { Y i ( τ ) } in the space M k (Γ g ( n )) such that the action of Γ g on the elements of the basis is described by a unitaryrepresentation ρ r of the finite Siegel modular group Γ g,n = Γ g / Γ g ( n ): Y i ( γτ ) = [det( Cτ + D )] k ρ r ( γ ) ij Y j ( τ ) , γ = A BC D ∈ Γ g . (2.17)6t variance with eq. (2.15), where only transformations of Γ g ( n ) were considered, in theprevious equation the full Siegel modular group Γ g is acting. Eq. (2.17) shows that the Siegelmodular forms { Y i ( τ ) } of given weight, level and genus have good transformation propertiesalso with respect to Γ g . To build a symplectic modular invariant supersymmetric theory, we have to specify the actionof Γ g on the matter multiplets ϕ of the theory, which can belong to separate sectors { ϕ ( I ) } .To this purpose we choose a particular level n . Both the genus g and the level n are keptfixed in the construction. The supermultiplets ϕ ( I ) of each sector I are assumed to transformin a representation ρ ( I ) of the finite Siegel modular group Γ g,n , with a weight k I . τ → γτ = ( Aτ + B )( Cτ + D ) − ,ϕ ( I ) → [det( Cτ + D )] k I ρ ( I ) ( γ ) ϕ ( I ) , γ = A BC D ∈ Γ g . (2.18)In the case of rigid supersymmetry, the action S of an N = 1 symplectic modular invariantsupersymmetric theory, restricted to Yukawa interactions, reads S = (cid:119) d xd θd ¯ θ K (Φ , ¯Φ) + (cid:119) d xd θ w (Φ) + h.c. , (2.19)and its invariance under eq. (2.18) requires the invariance of the superpotential w (Φ) and theinvariance of the Kahler potential up to a Kahler transformation: w (Φ) → w (Φ) K (Φ , ¯Φ) → K (Φ , ¯Φ) + f (Φ) + f ( ¯Φ) . (2.20)The invariance of the K¨ahler potential can be easily accomplished. A minimal K¨ahler poten-tial is: K = K τ + K ϕ , (2.21)where: K τ = − h Λ log det( − iτ + iτ † ) , h > , (2.22)with h a dimensionless constant and Λ some reference mass scale. K τ is invariant under thefull symplectic group Sp (2 g, R ) up to a K¨ahler transformation. The minimal K¨ahler potential K ϕ for matter multiplets ϕ ( I ) is invariant only under transformations of Γ g : K ϕ = (cid:88) I [det( − iτ + iτ † )] k I | ϕ ( I ) | . (2.23)The requirement of symplectic modular invariance for the superpotential is better appreciatedby expanding of w (Φ) in power series of the supermultiplets ϕ ( I ) : w (Φ) = (cid:88) n Y I ...I n ( τ ) ϕ ( I ) ...ϕ ( I n ) . (2.24) We restrict to integer modular weights. Fractional weights are in general allowed, but require a suitablemultiplier system [41, 73]. p -th order term to be modular invariant the functions Y I ...I p ( τ ) should transform asSiegel modular forms with weight k Y ( p ) in the representation ρ ( Y ) of Γ g,n : Y I ...I p ( γτ ) = [det( Cτ + D )] k Y ( p ) ρ ( Y ) ( γ ) Y I ...I p ( τ ) , (2.25)with k Y ( p ) and ρ ( Y ) such that:1. The weight k Y ( p ) should compensate the overall weight of the product ϕ ( I ) ...ϕ ( I p ) : k Y ( p ) + k I + .... + k I p = 0 . (2.26)2. The product ρ ( Y ) × ρ ( I ) × ... × ρ ( I p ) contains an invariant singlet.This framework can be easily extended to the case of local supersymmetry. Before discussing the inclusion of CP in this class of theories, we comment about a constraintapplying to their transformation laws under the Siegel modular group Γ g . The element S = − g commutes with all elements of Γ g , therefore by Schur’s Lemma the representationmatrix ρ r ( S ) of the finite modular group Γ g,n is proportional to a unit matrix, for anyrepresentation r . Furthermore, we have S = g and ρ r ( S ) = , which implies ρ r ( S ) = ± .Consider modular forms Y ( τ ) and matter multiplets ϕ at genus g and level n transformingin the same irreducible representation ρ r ( γ ) of Γ g,n under γ ∈ Γ g : Y ( γτ ) = [det( Cτ + D )] k Y ρ r ( γ ) Y ( τ ) , γ = A BC D ∈ Γ g , (2.27) ϕ γ −→ [det( Cτ + D )] k ϕ ρ r ( γ ) ϕ . (2.28)By choosing γ = S and observing that Y ( S τ ) = Y ( τ ), we get: Y ( τ ) = [det( − g )] k Y ρ r ( S ) Y ( τ ) . (2.29)If Y ( τ ) does not vanish, this implies:( − gk Y ρ r ( S ) = r . (2.30)Similarly, by considering twice the transformation S on the matter multiplet ϕ , we get: ϕ S −→ ( − gk ϕ ρ r ( S ) ϕ S −→ [( − gk ϕ ρ r ( S )] ϕ = ϕ , (2.31)and we find: ( − gk ϕ ρ r ( S ) = ± r . (2.32)The matrix ρ r ( S ) is independent from k , since it reflects a property of the group Γ g,n .Therefore, for any genus g , eqs. (2.30) and (2.32) provide a set of constraints on the weights k Y and k ϕ , once the representations ρ r is chosen. It is instructive to analyze the mutual8onsistency of eqs. (2.30) and (2.32). If the genus g is even, the only value of ρ r ( S ) compatiblewith eq. (2.30) is ρ r ( S ) = . Let R MF denote the set of all irreducible representations ofΓ g,n in (2.27) for all possible integer values of the weight k Y . This set might not coincidewith R , the set of all irreducible representations of Γ g,n . If R − R MF is not empty, there arerepresentations of Γ g,n , to which matter fields can be assigned, such that ρ r ( S ) = − . If thegenus g is odd, eq. (2.30) and eq. (2.32) become ( − k Y ρ r ( S ) = and ( − k ϕ ρ r ( S ) = ± ,respectively. We see that they can be compatible, provided the weights of the matter fields k ϕ satisfy: ( − k ϕ = ± ( − k Y . For example, let us consider g = 1 and the level n = 4.The inhomogeneous finite modular group is isomorphic to S , which has five irreduciblerepresentations , (cid:48) , , and (cid:48) . In the doublet representation ρ ( S ) = . Even weightfor matter fields in the doublet representation are possible, if we choose the plus sign ineq. (2.32). However, odd weights for matter multiplets in the doublet representation areallowed as well, and eq. (2.32) is satisfied with the minus sign. On the contrary, only evenweight modular forms can transform in the doublet representation.As we shall see, the relations (2.30) and (2.32) play an important role in the definition ofa consistent CP transformation law for matter multiplets and modular forms at genus g = 1and g = 2. In a theory invariant under Γ g = Sp (2 g, Z ), consistent CP transformations correspond toouter automorphism u ( γ ) of Γ g [46, 47]: CP γ CP − = u ( γ ) . (3.1)Each automorphism u ( γ ) of Γ g can be described by [74]: u ( γ ) = χ ( γ ) U γ U − , U ∈ Γ ∗ g , (3.2)where Γ ∗ g = GSp (2 g, Z ) denote the extended Siegel modular group, consisting of all integralmatrices U satisfying U t J U = ± J . The map χ ( γ ), called character of the Siegel modulargroup, is a homomorphism of Γ g into {± } . The group Γ ∗ g is generated by { S, T i , U } , wherethe matrix U , satisfying U J U − = − J , is defined as: U = − g g . (3.3)The Siegel modular group Γ g is a subgroup of Γ ∗ g and each element U of the group Γ ∗ g notbelonging to Γ g can be uniquely decomposed as: U = U γ (cid:48) γ (cid:48) ∈ Γ g . (3.4)Hence outer automorphisms u ( γ ) of Γ g are recovered from eq. (3.2) by replacing the genericelement U of Γ ∗ g either with the generator U or with the identity: u ( γ ) = χ ( γ ) U γ U − and u ( γ ) = χ ( γ ) γ . (3.5)Non-trivial characters χ ( γ ) exist only for g = 1 , g = 1 χ ( γ ) = { , (cid:15) ( γ ) } , where (cid:15) ( S ) = (cid:15) ( T ) = − . (3.6)ii) g = 2 χ ( γ ) = { , θ ( γ ) } , where θ ( S ) = 1 , θ ( T i ) = − . (3.7)iii) g ≥ χ ( γ ) = 1 . (3.8)For genus g = 1 , u , = χ , ( γ ) U γ U − ,satisfying the following relations: u = u = ( u u ) = . (3.9)Thus, the outer automorphism group is isomorphic to a Klein group K = { u , u , u = u u , u = } . For genus g > Z = { u, u = } . From eqs. (3.3) and (3.5) and the relation U γ U − = A − B − C D , (3.10)we can derive the action of the outer automorphisms on the generators of Γ g . g = 1 : u ( S ) = S − , u ( T ) = T − ,u ( S ) = − S − , u ( T ) = − T − ,u ( S ) = − S , u ( T ) = − T ,u ( S ) = S , u ( T ) = T , (3.11) g = 2 : u ( S ) = S − , u ( T i ) = T − i ,u ( S ) = S − , u ( T i ) = − T − i ,u ( S ) = S , u ( T i ) = − T i ,u ( S ) = S , u ( T i ) = T i , (3.12) g ≥ u ( S ) = S − , u ( T i ) = T − i ,u ( S ) = S , u ( T i ) = T i . (3.13) We denote by the identity element of the outer automorphism.
10e should select among these possibilities the good candidates to represent physical CPtransformations. Notice that the automorphism u is involutive for both u = u and u = u : u i ( u i ( γ )) = γ ( i = 1 , τ We assume that the CP transformation of moduli τ is linear and, for convenience, we write : τ CP −→ ( CP ) τ ≡ τ CP = X τ ∗ X t , (3.14)where X is an invertible g × g matrix such that Im ( τ CP ) >
0. The inverse CP transformationreads: τ CP − −−−→ τ CP − = ( X − τ ( X t ) − ) ∗ . (3.15)By enforcing the relation (3.1) on the generators S and T i of Γ g , for the automorphisms u a ( γ )( a = 1 , ..., XX t ( − τ − ) XX t = u a ( S ) τ ,τ + XB i X t = u a ( T i ) τ . (3.16)The elements u and u have the same action on τ and similarly for u , . As a consequenceeq. (3.16) reduces to: XX t ( − τ − ) XX t = − τ − , XB i X t = − ηB i , (3.17)where η = +1 for u , and η = − u , . Since B i form a set of basis of integral symmetricmatrices, we see that the second set of relations is solved by X = ± i g when u , andby X = ± g for u , . These solutions imply, respectively, XX t = − g and XX t = g ,both satisfying the first equation in (3.17). Since Im ( τ CP ) >
0, the only consistent choice is X = ± i g and we find: τ CP −→ τ CP = − τ ∗ , (3.18)as admissible CP transformation of the matrix τ . This represents the correct transformationlaw of the moduli for both u , outer automorphisms. We should instead discard u , . If wecombine eq. (3.18) with a modular transformation: τ γ −→ ( Aτ + B )( Cτ + D ) − CP −→ ( γ ◦ CP ) τ = ( − Aτ ∗ + B )( − Cτ ∗ + D ) − , (3.19)we get another allowed CP transformation. Since the theory is invariant under Γ g , this choiceshould not be view as independent from (3.18), which we take as representative element inthe class (3.19).By combining CP and the Siegel modular transformations we get the extended Siegelmodular group Γ ∗ g = GSp (2 g, Z ) and the full symmetry transformation of the complex moduliis τ → ( Aτ + B )( Cτ + D ) − for γ t J γ = J ,τ → ( Aτ ∗ + B )( Cτ ∗ + D ) − for γ t J γ = − J , (3.20) We see at the end of this Section that also non-linear actions of CP are allowed. They are howeverequivalent to the linear one. γ = A BC D and the CP transformation is represented by the matrix U in eq. (3.3).Notice that the action of γ and − γ on τ is the same and the full symmetry group acting onmoduli is isomorphic to P GSp (2 g, Z ) ≡ GSp (2 g, Z ) / {± g } . ϕ We consider a generic matter chiral supermultiplet ϕ , transforming as in eq. (2.28) under theSiegel modular group. We assume the following action of CP on ϕ : ϕ ( x ) CP −→ X r ϕ ( x P ) , (3.21)where X r is a unitary matrix, and a bar denotes hermitian conjugation. By realizing thecondition (3.1) on the matter field space, we find:[det( C ( τ ∗ ) CP − + D )] k ϕ X r ρ ∗ r ( γ ) X − r ϕ = χ ( γ ) gk ϕ [det( − Cτ + D )] k ϕ ρ r ( u ( γ )) ϕ (3.22)where we made use of the relation: u ( γ ) = χ ( γ ) A − B − C D . (3.23)We conclude that: X r ρ ∗ r ( γ ) X − r = χ ( γ ) gk ϕ ρ r ( u ( γ )) . (3.24)When the automorphism u is chosen, from the previous equation we find: X r ρ ∗ r ( S ) X − r = ρ r ( S − ) , X r ρ ∗ r ( T i ) X − r = ρ r ( T − i ) . (3.25)By making use of eqs. (3.6-3.7) and (3.11-3.12), for the automorphism u we get χ ( S ) gk ϕ ρ r ( u ( S )) = [( − gk ϕ ρ r ( S )] g ρ r ( S − ) χ ( T i ) gk ϕ ρ r ( u ( T i )) = [( − gk ϕ ρ r ( S )] ρ r ( T − i ) . (3.26)We exploit the results of Section 2.4 and discuss separately the cases g = 1 and g = 2. When g = 1, the matrix X r depends on both ρ r ( S ) and k ϕ and we distinguish two cases: • ( − k ϕ ρ r ( S ) = + r By combining eqs. (3.24) and (3.26), we get eq. (3.25) also for the automorphism u : X r ρ ∗ r ( S ) X − r = ρ r ( S − ) , X r ρ ∗ r ( T i ) X − r = ρ r ( T − i ) . (3.27) • ( − k ϕ ρ r ( S ) = − r In this case X r obeys: X r ρ ∗ r ( S ) X − r = − ρ r ( S − ) , X r ρ ∗ r ( T ) X − r = − ρ r ( T − ) . (3.28) We should more precisely denote X r as X r ( k ϕ ), but in the text we leave the dependence on k ϕ understood. r , with a suitablechoice of k ϕ .On the contrary, when g = 2, the representations r of the finite modular group fall intotwo classes: • ρ r ( S ) = + r By combining eqs. (3.24) and (3.26), we get again eq. (3.25): X r ρ ∗ r ( S ) X − r = ρ r ( S − ) , X r ρ ∗ r ( T i ) X − r = ρ r ( T − i ) . (3.29) • ρ r ( S ) = − r In this case X r obeys: X r ρ ∗ r ( S ) X − r = ρ r ( S − ) , X r ρ ∗ r ( T ) X − r = − ρ r ( T − ) . (3.30)Now X r is completely determined by ρ r ( S ) and does not depend on k ϕ .The automorphism u has been discussed in [50], for g = 1. The CP transformationdefined by eqs. (3.28) and (3.30) can possibly be consistently implemented if:1. the level n is even, which follows from taking the n − th power of the second relation ineqs. (3.28, 3.30).2. The dimension of representation ρ r is even, which follows from eqs. (3.28, 3.30) bytaking determinants.3. The traces of ρ r ( T ) and, for g = 1, ρ r ( S ) should vanish, which follows from eqs. (3.28,3.30) by taking traces.It is not inconceivable to build a model with these properties [50]. We do not deal with suchcase here and, in the rest of our paper, we discuss the automorphism u , focusing on thesolution ( − gk ϕ ρ r ( S ) = + r . Then, both u and u satisfy eq. (3.25), that can be regardedas a set of consistency conditions on the unitary matrix X r . Indeed, multiplying from theright each member of the previous equations by X r , we get linear equations in the unknown X r , which admits a unique solution, up to an overall phase factor.The action of CP in the moduli space, eq. (3.18), is involutive, that is CP τ = τ . This isnot necessarily the case in field space. From (3.1), by applying twice CP , we get: CP γ CP − = u ( u ( γ )) ≡ γ , (3.31)showing that CP is an inner automorphism, which can be induced by an element γ CP satisfying: γ CP γ γ − CP = γ . (3.32)Therefore γ CP belongs to the center of Γ g : { g , S } . By realizing the equality (3.31) in thematter field space, we obtain:det( Cτ + D ) k ϕ X r X ∗ r ρ r ( γ ) X − ∗ r X − r ϕ = det( Cτ + D ) k ϕ ρ r ( γ ) ϕ . (3.33)13y the Schur’s Lemma, the product X r X ∗ r is proportional to the identity. It acts withoutconjugating the matter fields and represents the element γ CP or, more precisely, the elementof the finite group Γ g,n that corresponds to γ CP . As a consequence we have: X r X ∗ r = r or X r X ∗ r = ρ r ( S ) = ± r . (3.34)The latter equality follows from S = g . Hence, the unitary matrix X r is either symmetricor antisymmetric. The indicator Ind r = | Γ g,n | (cid:80) γ ∈ Γ g,n Tr( ρ r ( γu ( γ ))) provides a criterion fordeciding whether X r X ∗ r = r or X r X ∗ r = − r [47]. The indicator Ind r is 1 and − X r X ∗ r = r , it is always possible to move to a basis where CP is canonical: X r = r .Then, the consistency conditions of eq. (3.25) imply that both ρ r ( S ) and ρ r ( T ) are symmetricin this basis: ρ t r ( S ) = ρ r ( S ) and ρ t r ( T i ) = ρ r ( T i ). Conversely, when ρ r ( S ) and ρ r ( T ) aresymmetric, the relations (3.25) unify into : X r ρ ∗ r ( γ ) X − r = ρ t r ( γ − ) , (3.35)which is solved by X r = r and the action of CP is involutive also in the field space. We seethat u ( γ ) lies in the same conjugacy class as γ − in the finite Siegel modular group. In otherwords u ( γ ) is a class-inverting automorphism of the finite Siegel modular group Γ g,n .When X r X ∗ r = ρ r ( S ) = − r , X r is a unitary antisymmetric matrix. In this case, thedimension of the representation ρ r has to be even. By performing a field redefinition we cango to a basis where X r takes the form [76] X r = iσ . . . iσ , (3.36)where σ is the second Pauli matrix. Examples of non-involutive CP transformations on thefield space have been given in ref. [77]. In most of our discussion, we focus on the involutivecase. Y ( τ ) Consider a multiplet Y ( τ ) of modular forms at genus g , level n and weight k Y , transformingas in eq. (2.27) under the Siegel modular group. In general, there can be several linearlyindependent such multiplets. We start by examining the case where there is only one. UnderCP it transforms as: Y ( τ ) CP −→ Y ( − τ ∗ ) . (3.37)We would like to establish the relation between Y ( − τ ∗ ) and X r Y ∗ ( τ ), where X r is the matrixspecifying the transformation law under CP of a matter multiplet ϕ at the same genus g ,level n , weight k ϕ = k Y and irreducible representation ρ r ( γ ), see eqs. (2.28) and (3.21). It follows from γ = S α · · · T β p i implying ρ t r ( γ − ) = ρ t r ( S − α ) . . . ρ t r ( T − β p i ) = ρ r ( S − α ) . . . ρ r ( T − β p i ) = ρ r ( S − α . . . T − β p i ) = ρ r ( u ( γ )) in a symmetric basis. X r satisfies necessarily the consistency condition associatedwith ( − gk ϕ ρ r ( S ) = + r . Indeed the other sign choice is only possible for r belonging to R − R MF , when g is even and k ϕ (cid:54) = k Y , when g is odd. Thus, among all possible matrices X r ,here the only relevant ones are the solution of the consistency conditions (3.25). We define: (cid:101) Y ( τ ) = X − ∗ r Y ∗ ( − τ ∗ ) . (3.38)We see that under γ ∈ Γ g , (cid:101) Y ( τ ) transforms as: (cid:101) Y ( γτ ) = X − ∗ r Y ∗ ( − ( γτ ) ∗ )= X − ∗ r Y ∗ ( u ( γ )( − τ ∗ ))= X − ∗ r [det( Cτ + D )] k χ ( γ ) gk ρ ∗ r ( u ( γ )) Y ∗ ( − τ ∗ )= [det( Cτ + D )] k ρ r ( γ ) (cid:101) Y ( τ ) , (3.39)where, in the last equality, we have used eq. (3.24). Since (cid:101) Y ( τ ) and Y ( τ ) transform in thesame way and, by assumption, there is only one linearly independent such modular form, weconclude that they are proportional, that is: Y ( − τ ∗ ) = λ X r Y ∗ ( τ ) . (3.40)By performing an additional CP transformation we have Y ( τ ) = | λ | X r X ∗ r Y ( τ ) = | λ | Y ( τ ),where we have made use of eq. (3.34). Note that only the solution X r X ∗ r = + r applies inthis case. The non vanishing constant λ can be absorbed by an appropriate choice of phaseof the whole multiplet Y ( τ ). In such a basis of modular forms Y ( τ ) we have: Y ( − τ ∗ ) = X r Y ∗ ( τ ) , (3.41)which reproduces the same CP transformation law of matter fields.If there are N linearly independent multiplets Y a ( τ ) ( a = 1 , ..., N ) transforming as in(2.27), eq. (3.39) holds individually for all (cid:101) Y a ( τ ) = X − ∗ r Y a ∗ ( − τ ∗ ) and we have: Y a ( − τ ∗ ) = λ ab X r Y b ∗ ( τ ) , (3.42)where λ ac λ ∗ cb X r X ∗ r = δ ab r . From eq. (3.34), now we can only deduce λ ac λ ∗ cb = ± δ ab , the signplus (minus) applying when the action of X r is (is not) involutive. When X r is involutive and λ ac λ ∗ cb = δ ab , it is always possible to factorize the matrix λ into λ = η − η ∗ and we obtain : η ab Y b ( − τ ∗ ) = X r [ η ab Y b ] ∗ ( τ ) . (3.43)We see that, by performing the change of basis Y a ( τ ) → η ab Y b ( τ ), eq. (3.41) holds indepen-dently for each multiplet Y a ( τ ). In applications we will use such basis of modular forms. If ( + λ ) is invertible, we take η = ( + λ ) − , then λη ∗− = η − . If ( + λ ) is not invertible, we canalways find a complex number u with | u | = 1 such that − u is not an eigenvalue of λ . Hence, λ + u isinvertible. In this case, we take η = ( u − λ + u ) − , then λη ∗− = η − . This construction was given by Prof.Marc van Leeuwen [78]. M k (Γ g ( n )) of weight k modular forms for Γ g ( n ) is finite dimensionaland decomposes into the sum of invariant subspaces, each carrying an irreducible represen-tation ( ρ a I r I ) i I j I of Γ g,n of dimension d I . Here a I is an index describing the degeneracy of therepresentation ρ r I and the indices ( i I , j I ) run from 1 to d I . Let { ( F a I r I ) i I ( τ ) } denote a basisin M k (Γ g ( n )). Our result (3.42) implies:( F a I r I ) i I ( − τ ∗ ) = δ IJ λ a I b J ( X r J ) j I i I ( F b J r J ) ∗ j I ( τ ) , (3.44)or, omitting indices, F ( − τ ∗ ) = X F F ∗ ( τ ) . (3.45)From the properties of λ ab and X r , it follows that X F X ∗ F = , X F ρ ∗ ( γ )( X F ) − = ρ ( u ( γ )) , (3.46)where ρ ( γ ) is the direct sum of the representations { ρ a I r I } .Actually, we could reverse the logic of this section and start by proving eqs. (3.45) and(3.46) and finally conclude that eq. (3.42) should hold. Indeed, for any element γ of Γ g ( n ),a basis { F ( τ ) } of M k (Γ g ( n )) obeys F ∗ ( − ( γτ ) ∗ ) = F ∗ ([ A ( − τ ∗ ) − B ][ − C ( − τ ∗ ) + D ] − ) = [det( Cτ + D )] k Y F ∗ ( − τ ∗ ) , (3.47)showing that F ∗ ( − τ ∗ ) belongs to M k (Γ g ( n )), which is the content of eq. (3.45). As a conse-quence, the following relation should be fulfilled: F ( τ ) = [ F ∗ ( − ( − τ ∗ ) ∗ )] ∗ = [ X ∗ F F ( − τ ∗ )] ∗ = X F X ∗ F F ( τ ) , (3.48)which leads to X F X ∗ F = . Finally, by performing a Siegel modular transformation γ ∈ Γ g on both sides of eq. (3.45) and using the identity − ( γτ ) ∗ = u ( γ )( − τ ∗ ), we obtain X F ρ ∗ ( γ )( X F ) − = χ ( γ ) gk Y ρ ( u ( γ )) = ρ ( u ( γ )) , (3.49)where ρ ( γ ) = diag( ρ r ( γ ) , . . . , ρ m r ( γ ) , . . . , ρ r p ( γ ) , . . . , ρ mp r p ( γ )) is generally reducible. Theneq. (3.42) follows by projecting (3.45) on invariant subspaces. We have seen that the transformation properties of moduli, matter multiplets and modularforms are given by: τ CP −→ − τ ∗ ,ϕ ( x ) CP −→ X r ϕ ( x P ) ,Y a ( τ ) CP −→ Y a ( − τ ∗ ) = λ ab X r Y b ∗ ( τ ) , (3.50)with λ ac λ ∗ cb X r X ∗ r = δ ab r . For both automorphism u and the positive branch ( − gk ϕ ρ r ( S ) =+ r of u , the unitary matrix X r solves the consistency conditions (3.25) and is determinedup to an arbitrary phase. When the action of CP on the field space is involutive, it is conve-nient to move to the basis where X r = . Here the matrices ρ r ( S ) and ρ r ( T ) are symmetric16nd it is possible to work with modular forms Y a ( τ ) where λ ab = δ ab . The minimal K¨ahlerpotential, eqs. (2.22) and (2.23), is always invariant under CP. For the superpotential, thecondition for CP invariance simplifies when i) X r = , ii) the Clebsh-Gordan coefficientsare all real in the adopted basis and iii) the modular forms Y ( τ ) are in a basis where (3.41)holds. In this case the superpotential is CP invariant when its free parameters are real. There are points of the moduli space H g where CP is conserved. The theory is invariant underΓ g . Moreover the inequalities (2.11) defining the fundamental domain F g do not change whenwe map τ into − τ ∗ . Thus it is sufficient to look for the CP invariant points belonging to F g .If τ belongs to the interior of F g , the CP invariant points are the solutions of: − τ ∗ = τ , (4.1)that is the moduli τ of the fundamental domain with vanishing real part. Points of theboundary ∂ F g of F g , where at least one of the relations in eq. (2.11) is realized as an equality,are related by modular transformations. The requirement of CP conservation for any point τ of the boundary is the existence of a modular transformation γ , such that: − τ ∗ = γτ . (4.2)Indeed the composition of a CP transformation with a modular transformation is an equiv-alent CP transformation in our theory. By applying two consecutive such combinations, weobtain the condition: u ( γ ) γτ = τ . (4.3)We make use of the identity [72]( Cτ ∗ + D ) t Im ( γτ )( Cτ + D ) = Im ( τ ) . (4.4)If τ satisfies eq. (4.2), we have Im ( γτ ) = Im ( − τ ∗ ) = Im ( τ ) and( Cτ ∗ + D ) t Im ( τ )( Cτ + D ) = Im ( τ ) . (4.5)By taking the determinant of both sides and noticing Im ( τ ) >
0, we get | det( Cτ + D ) | = 1 . (4.6)By comparing eq. (4.6) with eq. (2.11), we see that indeed τ belongs to the boundary ∂ F g .We can distinguish two cases where eq. (4.6) is satisfied: C = 0 and C (cid:54) = 0. The elements ofΓ g having C = 0 are of the type: γ = A M A t − (4.7)17here A is an unimodular integral matrix and M is a symmetric modular matrix. In thiscase the relation (4.2) becomes: − Re ( τ ) = A ( Re ( τ ) + M ) A t Im ( τ ) = A Im ( τ ) A t . (4.8)If we consider A = g , we get: 2 Re ( τ ) + M = 0. Since | Re ( τ ) | ≤ /
2, this relation is solvedby: Re ( τ ) = 0 , and Re ( τ ij ) = ±
12 (4.9)for M = g and M = ± B i respectively. If C (cid:54) = 0, we consider as an example the choice γ = S . The condition eq. (4.2) becomes: τ ∗ = τ − . (4.10)For genus g = 1, this is the arc | τ | = 1. In the case of g = 2, the CP conserved valuesof τ satisfy the conditions | τ | + | τ | = | τ | + | τ | = 1, τ τ ∗ + τ ∗ τ = 0, which are | τ | = | τ | = 1, τ = 0 in the fundamental domain F . We might ask whether all pointsof the boundary ∂ F g ( g ≥
2) enjoy CP invariance, as is the case for genus one. All pointsof ∂ F g satisfying | Re ( τ ij ) | = 1 /
2, for at least one pair ( ij ), are easily shown to be CPconserving, since the relation (4.2) is satisfied by a translation γ . For points satisfying thecondition | det( Cτ + D ) | = 1 with C (cid:54) = 0, it is more difficult to establish, in general, if theyare CP conserving or not. Showing this amounts to prove that, for a generic τ satisfying | det( Cτ + D ) | = 1 with C (cid:54) = 0, we can always determine a modular transformation γ suchthat eq. (4.2) holds. To our best knowledge this is still an open problem.In summary, in the interior of the F g , a point τ is CP conserving if and only if its realpart is vanishing. There are additional CP conserving points on the boundary of F g , but wewere not able to prove that any point belonging to ∂ F g is CP conserving. Consider a point τ of the fundamental domain where CP is conserved. Then there is anelement γ of Γ g such that − τ ∗ = γτ . As we have seen, eq. (4.6), the combination D = det ( Cτ + D ) (4.11)is a pure phase. Assume that the lepton sector is described by the superpotential: W = − E ci Y eij ( τ ) L j H d − L i Y νij ( τ ) L j H u H u , (4.12)where under the group element γ the matter multiplets ϕ ( ϕ = H u,d , E c , L ) transform as: ϕ γ −→ D k ϕ ρ ϕ ( γ ) ϕ . (4.13) ( | det A | = 1). k E c ,L carry a flavour index and D k Ec,L are diagonal unitary matrices in flavourspace. The invariance of the W under the modular transformation γ implies: Y e ( γτ ) = D − k Hd ρ † H d ( γ ) D − k Ec ρ ∗ E c ( γ ) Y e ( τ ) ρ † L ( γ ) D − k L , Y ν ( γτ ) = D − k Hu ρ † H u ( γ ) D − k L ρ ∗ L ( γ ) Y ν ( τ ) ρ † L ( γ ) D − k L , (4.14)On the other hand, the invariance of W under a CP transformation requires: Y e ( − τ ∗ ) = X † d X ∗ E c Y e ∗ ( τ ) X † L , Y ν ( − τ ∗ ) = X † u X ∗ L Y ν ∗ ( τ ) X † L . (4.15)By combining eqs. (4.14) and (4.15), at the point τ enjoying residual CP symmetry we get: Y e ∗ ( τ ) = Ω d Ω TE c Y e ( τ ) Ω L , Y ν ∗ ( τ ) = Ω u Ω TL Y ν ( τ ) Ω L , (4.16)where we have defined unitary matrices:Ω ϕ = ρ † ϕ ( γ ) D − k ϕ X ϕ , ( ϕ = H u,d , E c , L ) . (4.17)The charged lepton and neutrino mass matrices are given by M e = Y e v d and M ν = Y ν v u / Λfor the minimal K¨ahler potential (2.23). Thus at the point τ enjoying residual CP invariancewe have M † e ( τ ) M e ( τ ) = Ω † L (cid:2) M † e ( τ ) M e ( τ ) (cid:3) ∗ Ω L ,M † ν ( τ ) M ν ( τ ) = Ω † L (cid:2) M † ν ( τ ) M ν ( τ ) (cid:3) ∗ Ω L . (4.18)We see that the hermitian combination of the neutrino and charged lepton mass matrices M † ν M ν and M † e M e are invariant under a common transformation of the left-handed chargedleptons and the left-handed neutrinos, which represents a combination of CP and modulartransformations. The unitary matrix Ω L should be symmetric otherwise the neutrino andthe charged lepton mass spectrum would be constrained to be partially degenerate [45].Furthermore, the conditions eq. (4.18) fulfilled at the CP fixed point imply that both Diracand Majorana CP phases are trivial [45]. Therefore values of moduli deviating from residualCP symmetry fixed points are required to accommodate the observed non-degenerate leptonmasses and a non-trivial Dirac CP. The supersymmetric modular invariant theory of Section 2 can be consistently defined evenwhen τ belongs to an invariant subspace Ω of the Siegel moduli space H g . In this case thefull Siegel modular group Γ g is replaced by a convenient subgroup N ( H ). The points τ in Ωare fixed points of the subgroup H of Γ g [79–81], while N ( H ), the normalizer of H , leaves Ωinvariant as a whole. The elements γ of N ( H ) satisfy the relation γ − Hγ = H . Depending on H , the complex dimensionality of the invariant subspace Ω can range from zero to g ( g + 1) / τ , since we are not guaranteed that − τ ∗ belongs to Ω. To remedy this situation, we canadopt the definition of CP given in eq. (3.19), where a modular transformation γ is followedby the canonical CP transformation τ → ( γ ◦ CP ) τ = ( − Aτ ∗ + B )( − Cτ ∗ + D ) − , (5.1)and require for any τ ∈ Ω: ( γ ◦ CP ) τ = τ (cid:48) ∈ Ω . (5.2)Since each point of Ω is fixed under H , the transformation ( γ ◦ CP ) should obey:( γ ◦ CP ) ◦ H = H ◦ ( γ ◦ CP ) . (5.3)This condition determines γ up to a modular transformation γ (cid:48) of the normalizer N ( H ),given that it is satisfied by both ( γ ◦ CP ) and ( γ (cid:48) γ ◦ CP ). Therefore, it is sufficient to choosea representative CP transformation in this class, and we denote it as CP s . For the caseof g = 2, all independent invariant subspaces and the corresponding CP transformationsare summarized in table 1, where all CP s are chosen to be involutive with ( CP s ) τ = τ .There are 6 zero-dimensional invariant subspaces. The unique point belonging to each ofthese regions is CP invariant. In the modular subspaces of complex dimension one and twothere are infinite CP-conserving points. We list below the CP-conserving points satisfying( CP s ) τ = g − i τ , where g i stand for the generators of N ( H ). This request can be equivalentlycast in the form : ( g i ◦ CP s ) τ = τ . (5.4)1. τ τ : G ◦ CP s : Re ( τ ) = 0 , | τ | = 1 ,G (cid:48) ◦ CP s : Re ( τ ) = 0 , | τ | = 1 ,G ◦ CP s : | Re ( τ ) | = 1 / , Re ( τ ) = 0 ,G (cid:48) ◦ CP s : | Re ( τ ) | = 1 / , Re ( τ ) = 0 ,G ◦ CP s : Re ( τ ) = − Re ( τ ) , Im ( τ ) = Im ( τ ) . τ τ τ τ : G ◦ CP s : | Re ( τ ) | = 1 / , Re ( τ ) = 0 ,G ◦ CP s : | Re ( τ ) | = 1 / , Re ( τ ) = 0 ,G ◦ CP s : | τ | = 1 , τ = 0 ,G ◦ CP s : Re ( τ ) = Im ( τ ) = 0 . Given any CP conserving point τ of g i ◦ CP s , ( CP s ) τ would be the fixed point of g − i ◦ CP s . ixed points τ CP s Fixed points τ CP s τ τ CP τ τ τ τ CP i τ CP ω τ T − ◦ CP τ τ CP τ / / τ T ◦ CP τ τ / τ / τ CP ζ ζ + ζ − ζ + ζ − − ζ − (( ST ) T ) − ◦ CP η ( η − ( η − η ST ( T T ) − S ◦ CP i i CP ω ω ( T T ) − ◦ CP i √ CP ω i T − ◦ CP — Table 1:
The generalized CP transformation in the modular subspace in Siegel upper half plane H . The complex moduli are denoted by τ , τ , τ , and ζ = e πi/ , η = (1 + i √ , ω = e πi/ . i τ : R ◦ CP s : Re ( τ ) = 0 ,G ◦ CP s : | τ | = 1 ,G ◦ CP s : | Re ( τ ) | = 1 / . ω τ : R ◦ CP s : Re ( τ ) = 0 ,G ◦ CP s : | τ | = 1 ,G ◦ CP s : | Re ( τ ) | = 1 / . τ τ : R ◦ CP s : Re ( τ ) = 0 ,G ◦ CP s : | τ | = 1 ,G ◦ CP s : | Re ( τ ) | = 1 / . τ / / τ : G ◦ CP s : Re ( τ ) = 0 ,R ◦ CP s : | Re ( τ ) | = 1 / ,G R ◦ CP s : ( Re ( τ ) ± / + ( Im ( τ )) = 1 / . τ τ / τ / τ : G ◦ CP s : | τ | = 2 / √ ,G ◦ CP s : ( Re ( τ ) ± / + ( Im ( τ )) = 4 / ,G (cid:48) ◦ CP s : Re ( τ ) = 0 . Here G , G , G (cid:48) , G (cid:48) , R in each case are the generators of N ( H ) associated with the twobidimensional and the five unidimensional invariant subspaces shown in table 1. Their matrixrepresentation in a convenient basis is given in ref. [2]. In a CP-invariant theory, CP violation can arise only as a consequence of the choice of thevacuum. If in addition the theory enjoys flavour modular invariance, the properties of theobserved fermion spectrum such as masses, mixing angles and phases could be determinedmostly by the vacuum, rather than by Lagrangian parameters. These features are parts ofan appealing framework for the unification of flavor, CP and modular symmetries, advocatedin recent works [39–44] in a top-down perspective. In a bottom-up approach we can hope toexplore some aspect of this ideal framework.In this spirit, here we present an example of how CP and flavour symmetries can becombined and enforced in a supersymmetric theory, where the properties of the mass spectrumdepend in a non-trivial way on a multidimensional moduli space. When a single modulusexists, analogous studies have been carried out in refs. [48, 50–53]. We focus on the leptonsector, where we have 12 relevant observables. We show that all measured mass combinationsand mixings can be correctly described in terms of five Lagrangian parameters and by aconvenient point in moduli space . The theory is CP invariant and all 3 CP violatingphases arise spontaneously as a consequence of a small departure of τ from a CP-symmetricpoint.It is convenient to base our construction on the invariant subspace τ = τ at genus g = 2 [2]. At level 2, the modular group N ( H ) is S × Z , whose generators G = T T , G = T and G = S are shown in appendix A, in a convenient basis. We adopt u ( γ ) = u ( γ )as the automorphism defining CP. In terms of G i ( i = 1 , , X r ρ ∗ r ( G i ) X − r = ρ r ( u ( G i )) = ρ r ( G − i ) , (6.1) Note that the most economic modular-invariant flavour models describe the lepton sector by making useof five parameters. With an abuse of language we keep denoting by N ( H ) also the projection of N ( H ) at level two. u ( G ) = u ( T T ) = T − T − = T − T − = G − . Since in theadopted basis all the generators G , G and G are represented by unitary and symmetricmatrices, we can choose the canonical CP, X r = r . Moreover, the Clebsh-Gordan coefficientsare all real in this basis, and we will work with modular forms Y ( τ ) for which eq. (3.41) holds.By choosing minimal kinetic terms, we conclude that the theory is CP invariant when allLagrangian parameters are real.We choose the same field content and the same weight and representation assignment asin Lepton model II of ref. [2]. The matter chiral multiplets consist of three SU (2) L singlets E c , three doublets L , two Higgs doublets H u,d , transforming as ρ E c = ⊕ , ρ L = (cid:48) , ρ H u = ρ H d = ,k H u = k H d = 0 , k E cD = − , k E c = k L = − . (6.2)Neutrino masses are described by the Weinberg operator. The superpotential of the leptonsector includes: w e = α ( E cD LY (4) (cid:48) a ) H d + β ( E cD LY (4) (cid:48) b ) H d + γ ( E c LY (cid:48) ) H d ,w ν = g Λ (
LLY (cid:48) ) H u H u + g Λ (
LLY ) H u H u . (6.3)The phases of parameters α, γ and g are unphysical and can be always removed, even beforeenforcing CP invariance.We impose CP invariance, requiring also β and g to be real. At this level, the predictionsof the model depend on five Lagrangian parameters plus the complex values of τ and τ .From the superpotential and from the Clebsh-Gordan coefficients of S × Z , the chargedlepton and neutrino mass matrices read: M e = (cid:32) α ( √ Y (4) (cid:48) a, − Y (4) (cid:48) a, )+ β ( √ Y (4) (cid:48) b, − Y (4) (cid:48) b, ) √ αY (4) (cid:48) a, + √ βY (4) (cid:48) b, − αY (4) (cid:48) a, − βY (4) (cid:48) b, −√ αY (4) (cid:48) a, −√ βY (4) (cid:48) b, α ( √ Y (4) (cid:48) a, +2 Y (4) (cid:48) a, )+ β ( √ Y (4) (cid:48) b, +2 Y (4) (cid:48) b, ) 2 αY (4) (cid:48) a, +2 βY (4) (cid:48) b, γY γY γY (cid:33) v d ,M ν = g ( √ Y + Y ) + g Y √ g Y g Y √ g Y g ( −√ Y + Y ) + g Y g Y g Y g Y − g Y + g Y v u Λ . (6.4)Here Y (cid:48) = ( Y , Y , Y ) and Y = Y are weight 2 modular forms, while Y (4) (cid:48) a = ( Y (4) (cid:48) a, , Y (4) (cid:48) a, , Y (4) (cid:48) a, )and Y (4) (cid:48) b = ( Y (4) (cid:48) b, , Y (4) (cid:48) b, , Y (4) (cid:48) b, ) are weight 4 modular forms. Their explicit expressions interms of the second order theta constants are given in appendix B. We find a good agree-ment between the model predictions and the experimental data, for the following choice ofparameters: τ = − . . i , τ = − . . i ,β/α = − . , γ/α = 0 . , g /g = 1 . ,αv d = 39 . , g v u / Λ = 6 . . (6.5)23arameters Best fit value and 1 σ error m e /m µ . ± . m µ /m τ . ± . m / − eV . +0 . − . ∆ m / − eV . +0 . − . δ CP /π . +0 . − . sin θ . +0 . − . sin θ . +0 . − . sin θ . +0 . − . Table 2:
The best fit values and the 1 σ ranges of the charged lepton mass ratios and the leptonmixing parameters. The charged lepton mass ratios averaged over tan β [5] are taken from ref. [82],and we adopt the values of the lepton mixing parameters from NuFIT v5.0 with Super-Kamiokandaatmospheric data for normal ordering [83]. The experimental 1 σ ranges shown in table 2, with the exclusion of that referring to theDirac CP phase δ CP /π , are the input data in our fit. Accordingly, the lepton masses andmixing parameters are determined to be:sin θ = 0 . , sin θ = 0 . , sin θ = 0 . , δ CP = 1 . π ,α = 0 . π , α = 1 . π , m e /m µ = 0 . , m µ /m τ = 0 . ,m = 9 .
22 meV , m = 12 .
62 meV , m = 52 .
07 meV ,m β = 12 .
77 meV , m ββ = 10 .
74 meV , (6.6) m β and m ββ being the effective neutrino masses in beta decay and neutrinoless double betadecay, respectively. Note that the model predicts a Dirac CP phase δ CP close to 3 π/
2. Theneutrino masses are of normal hierarchy type and they are quite tiny. All the experimentalbounds from neutrino oscillations [83], tritium beta decays [84], neutrinoless double decay [85]and cosmology [86] are satisfied. We can make a further step and restrict the complex modulito the one-dimensional invariant subspace τ = τ / τ / τ = τ .We find that the observed lepton masses and mixing angles can still be accommodated .The best fit values of the remaining input parameters are given by: τ = − . . i , ( τ = τ = 2 τ ) β/α = − . , γ/α = 0 . , g /g = 1 . ,αv d = 38 . , g v u / Λ = 6 . . (6.7) Notice that another possible one-dimensional subspace, τ = 0, can also fit the data well, except thatsin θ is slightly smaller than the 3 σ lower bound of the experimental value. igure 1: The correlations among the input free parameters, neutrino mixing angles and CPviolating phases in the model discussed in the text, where the moduli τ are restricted to the subspacewith τ = τ = 2 τ . The zero-dimensional fixed point τ = i √ ( ) is marked by pink triangle inthe τ plane . The masses and mixing parameters of leptons are predicted to be:sin θ = 0 . , sin θ = 0 . , sin θ = 0 . , δ CP = 1 . π ,α = 0 . π , α = 1 . π , m e /m µ = 0 . , m µ /m τ = 0 . ,m = 10 .
08 meV , m = 13 .
26 meV , m = 51 .
26 meV ,m β = 13 .
40 meV , m ββ = 11 .
26 meV . (6.8)We have comprehensively explored the parameter space of the model, within the invariant25ubspace τ = τ = 2 τ . Requiring the three lepton mixing angles and neutrino squaredmass splittings ∆ m , ∆ m to lie in the experimentally allowed 3 σ regions [83], we get thecorrelations between the free parameters and observable quantities shown in figure 1. It isworth noting that τ = − . . i is close to 2 i/ √ ≈ . i and the VEVs ofmoduli that best reproduce the data are all clustered near the zero-dimensional fixed point i √ ( ). This point preserves CP and a Z residual symmetry generated by ( ST ) T SV . Wesee that a small deviation from the CP-conserving point is sufficient to generate a sizableamount of CP violation.
Figure 2:
The distribution of CP violation measured by the average of the three CP phases ∆ CP in the subspace with τ = τ , τ = 0 (left panel) and τ = τ = 2 τ (respectively right panel). Inthe subspace with τ = 0, | Re ( τ ) | = 0 , / | τ | = 1 are CP-conserving points. In the subspacewith τ = τ / | Re ( τ ) | = 0 and | τ | = 2 / √ (cid:0) i i (cid:1) , ( ω ω ) and i √ ( ) are marked by pink dot, hexagonal star and triangle, respectively. In this model we can look numerically for the points preserving CP and compare themwith those discussed analytically in the previous Section. By varying the value of τ in themodular subspaces with τ = τ , τ = 0 or τ = τ = 2 τ , while keeping all other parametersfixed to their best fit values, we present the CP violating quantity ∆ CP ≡ ( | sin δ CP | + | sin α | + | sin α | ) as two heatmap plots in figure 2. The colors of the points in the figurechange from blue to red, indicating that the ∆ CP is increasing: ∆ CP = 0 corresponds toCP conservation and ∆ CP = 1 corresponds to maximal CP violation. The CP conservedpoints (shown with dark blue colour) are indeed consistent with our analytic results. It isremarkable that significant CP violation can be induced for small deviation of τ from thezero-dimensional fixed points ( i i ), ( ω ω ) and i √ ( ). CP symmetry and its violation are key elements of a correct description of particle inter-actions. They are also crucial in explaining the baryonic asymmetry of the universe. CP26iolation has been observed in a rich variety of physical processes, but it is traceable to aunique source: a single observable phase in the CKM mixing matrix. A similar, yet-to-be-discovered, source can reside in the lepton mixing matrix, thus closely linking the asymmetrybetween the properties of particles and antiparticles to the features of the fermionic mass spec-trum. CP transformations are a basic ingredient of any description of particle interactions.In theories invariant under the action of a local, continuous, gauge group, the existence of CPtransformations, inverting the sign of commuting gauge charges is always guaranteed [87].In general, up to topological terms, pure gauge interactions are automatically CP invariant,while Yukawa interactions are not. Nonetheless, the possibility that CP is a symmetry ofthe entire theory, including the Yukawa sector, is very appealing. The observed degree ofCP violation would arise as a consequence of the choice of the vacuum and not from theadjustment of ad-hoc free parameters.There is an interesting class of flavour models where the vacuum plays a key role inthe description of fermion masses and mixing angles. Here both the flavour group and theSB sector have a common root. SB is parametrized by moduli, scalars taking values in asymmetric space of the type
G/K . A discrete, modular subgroup Γ of G , acting on G/K ,plays the role of flavour symmetry. The symmetry associated with Γ is a gauge symmetryand is related to the redundancy of the vacuum description. Physically inequivalent vacuaare described by a fundamental domain (
G/K ) / Γ. Thus the vacuum is specified by a pointin the multidimensional moduli space, up to a discrete modular transformation. To preservethe structure of (
G/K ) / Γ, CP transformations are to be searched among the nontrivialautomorphisms of Γ. The existence of such automorphisms allows to enforce CP invarianceand to provide a common origin of fermion masses, mixing angles and CP violating phases.Pursuing a bottom-up approach, we have analyzed the allowed CP definitions in symplec-tic modular invariant theories, where G = Sp (2 g, R ) and K = U ( g ), starting from a completeclassification of the automorphisms of the symplectic modular group Γ = Sp (2 g, Z ). Noticethat the symplectic modular group Sp (2 g, Z ) coincides with SL (2 , Z ) for the smallest genus g = 1. A unique possibility emerges when g ≥
3, while two are allowed for g ≤
2. We havealso discussed the action of CP transformations on moduli, matter multiplets and modularforms, the building blocks for the construction of flavour models. In these theories, physicallyinequivalent vacua are described by a fundamental domain F g in the Siegel upper half plane,whose explicit construction is known only at genus one and two. We have shown that in theinterior of F g CP is preserved only on the surface Re ( τ ) = 0, while on its boundary there areinfinite CP-conserving points. An interesting open problem is to establish whether all thepoints of the boundary are CP-conserving, like in genus one, or not.Finally, we have shown how to combine all the previously discussed elements in the con-struction of a CP and symplectic invariant model of lepton masses at genus two. In theadopted framework, where the K¨ahler potential is minimal, the representations of the finitemodular group are symmetric, the Clebsh-Gordan coefficients are all real and a suitable basisof modular forms is chosen, CP invariance is enforced by requiring that all Lagrangian pa-rameters are real. In our model we manage to correctly reproduce the observed lepton massesand mixing angles by using five real free parameters. Neutrinos are Majorana particles, witha normally ordered mass spectrum. The model predicts all the three CP-violating phases,with the value of δ CP approaching 3 π/
2. 27n our analysis, we have not attempt to determine dynamically the vacuum [88–90].Rather, we have treated the moduli VEV as additional free parameters, optimized to max-imize the agreement between data and predictions, with the hope of gaining some insightinto the nature of the preferred vacuum. It is remarkable that the best values of moduliobtained in this way are very close to a point of enhanced symmetry, where both CP andsome finite modular transformations are preserved. Thus, it suffices a small departure from aCP-conserving vacuum to generate sizable CP-violating effects. This confirms an intriguingbehaviour already noticed in genus one constructions [91–93], where also the charged fermionhierarchy can benefit from the proximity to one such vacuum [94, 95]. If the previously out-lined scenario is acceptable and provides a good enough description of the real world, we areconfronted with a fascinating question: why is our universe living so close to a critical point?
Appendices
A The finite Siegel modular group S × Z S × Z can be generated by three generators: G ≡ T T , G ≡ T , G ≡ S satisfying the multiplication rules: G = G = G = ( G G ) = ( G G ) = ( G G G ) = 1 . (A.1)The S and Z subgroups are generated by S = G , T = ( G G ) and V = ( G G ) respec-tively, which obey the relations: S = T = ( ST ) = 1 , V = 1 , SV = VS , T V = VT . (A.2)The generators G , , can be expressed in terms of S , T and V as G = S , G = (( ST ) T S ) T V , G = ( ST ) T S .The group S × Z has four singlet representations , (cid:48) , ˆ1 , ˆ1 (cid:48) , two doublet representations , ˆ2 , and four triplet representations , (cid:48) , ˆ3 and ˆ3 (cid:48) . When constructing a CP and symplecticmodular invariant model, it is more convenient to work in the basis of X r = r . Since theindicator Ind r = +1 in all representations r , such a basis can really be achieved. For thesinglet representations, we have ( ˆ1 ) : S = T = 1 , V = 1 ( − , (cid:48) ( ˆ1 (cid:48) ) : S = − , T = 1 , V = 1 ( − , (A.3)In the doublet representations, the generators are represented by ( ˆ2 ) : S = 12 (cid:18) −√ −√ − (cid:19) , T = 12 (cid:18) − √ −√ − (cid:19) , V = ( − ) . (A.4)28or the doublet representations, the generators are ( ˆ3 ) : S = 16 − √ √ √ − √ √ √ , T = 12 − −√ √ − , V = ( − ) , (cid:48) ( ˆ3 (cid:48) ) : S = − − √ √ √ − √ √ √ , T = 12 − −√ √ − , V = ( − ) . (A.5)It is easy to check that the representation matrices of G , G and G are unitary and sym-metric in all irreducible representations. As a consequence, the CP symmetry for the auto-morphism u is exactly the canonical CP in this basis with X r = r , as shown in Section 6.The decomposition rules of the Kronecker product of two irreducible representations arenecessary in model construction. We report the Kronecker products and the Clebsch-Gordancoefficients in the above CP basis in table 3. The notations α i and β i refer to the elementsof the first and the second representation of the product respectively. B Siegel modular forms of genus g = 2 at level n = 2 There are five linearly independent Seigel modular forms p , , , , at weight k = 2 and level n = 2, and they are form a quintet of the finite modular group Γ , ∼ = S [96]: p = Θ[00] ( τ ) + Θ[01] ( τ ) + Θ[10] ( τ ) + Θ[11] ( τ ) ,p = 2 (cid:0) Θ[00] ( τ )Θ[01] ( τ ) + Θ[10] ( τ )Θ[11] ( τ ) (cid:1) ,p = 2 (cid:0) Θ[00] ( τ )Θ[10] ( τ ) + Θ[01] ( τ )Θ[11] ( τ ) (cid:1) ,p = 2 (cid:0) Θ[00] ( τ )Θ[11] ( τ ) + Θ[01] ( τ )Θ[10] ( τ ) (cid:1) ,p = 4Θ[00]( τ )Θ[01]( τ )Θ[10]( τ )Θ[11]( τ ) . (B.1)where Θ is the second order theta constant defined by:Θ[ σ ]( τ ) = (cid:88) m ∈ Z g e πi ( m + σ/ τ ( m + σ/ t , (B.2)where σ = ( σ , σ , . . . , σ g ) are row vectors with σ i = 0 ,
1. When we restrict τ to the two-dimensional modular subspace with τ = τ , the relation p ( τ ) = p ( τ ) is fulfilled, thusthe modular forms space of weight 2 collapses into a four-dimensional subspace. The Siegelmodular forms of weight 2 and level 2 can be arranged into a singlet and a triplet of the finiteSiegel modular subgroup S × Z : (cid:48) : Y (cid:48) ( τ ) = √ p ( τ ) − p ( τ ) − p ( τ ) − p ( τ )) p ( τ ) − p ( τ ) − p ( τ ) + 6 p ( τ ) √ − p ( τ ) − p ( τ ) + p ( τ )) ≡ Y ( τ ) Y ( τ ) Y ( τ ) , : Y ( τ ) = p ( τ ) + 3 p ( τ ) ≡ Y ( τ ) . (B.3)29 ⊗ = ˆ1 ⊗ ˆ2 = , ⊗ ˆ2 = ˆ1 ⊗ = ˆ2 1 (cid:48) ⊗ = ˆ1 (cid:48) ⊗ ˆ2 = , (cid:48) ⊗ ˆ2 = ˆ1 (cid:48) ⊗ = ˆ22 , ˆ2 ∼ (cid:18) αβ αβ (cid:19) , ˆ2 ∼ (cid:18) αβ − αβ (cid:19) ⊗ = (cid:48) ⊗ (cid:48) = ˆ1 ⊗ ˆ3 = ˆ1 (cid:48) ⊗ ˆ3 (cid:48) = , ⊗ (cid:48) = (cid:48) ⊗ = ˆ1 ⊗ ˆ3 (cid:48) = ˆ1 (cid:48) ⊗ ˆ3 = (cid:48) , ⊗ ˆ3 = (cid:48) ⊗ ˆ3 (cid:48) = ˆ1 ⊗ = ˆ1 (cid:48) ⊗ (cid:48) = ˆ3 1 ⊗ ˆ3 (cid:48) = (cid:48) ⊗ ˆ3 = ˆ1 ⊗ (cid:48) = ˆ1 (cid:48) ⊗ = ˆ3 (cid:48) , ˆ3 ∼ αβ αβ αβ (cid:48) , ˆ3 (cid:48) ∼ αβ αβ αβ ⊗ = ˆ2 ⊗ ˆ2 = s ⊕ (cid:48) a ⊕ s , ⊗ ˆ2 = ˆ1 ⊕ ˆ1 (cid:48) ⊕ ˆ21 s , ˆ1 ∼ α β + α β (cid:48) a , ˆ1 (cid:48) ∼ α β − α β s , ˆ2 ∼ (cid:18) α β + α β α β − α β (cid:19) ⊗ = ˆ2 ⊗ ˆ3 = ⊕ (cid:48) , ⊗ ˆ3 = ˆ2 ⊗ = ˆ3 ⊕ ˆ3 (cid:48) ⊗ (cid:48) = ˆ2 ⊗ ˆ3 (cid:48) = ⊕ (cid:48) , ⊗ ˆ3 (cid:48) = ˆ2 ⊗ (cid:48) = ˆ3 ⊕ ˆ3 (cid:48) , ˆ3 ∼ √ α β − √ α β − α β √ α β + √ α β + 2 α β α β − α β (cid:48) , ˆ3 (cid:48) ∼ −√ α β − √ α β + 2 α β ) −√ α β + √ α β + 2 α β )2 α β + 2 α β , ˆ3 ∼ −√ α β − √ α β + 2 α β ) −√ α β + √ α β + 2 α β )2 α β + 2 α β (cid:48) , ˆ3 (cid:48) ∼ √ α β − √ α β − α β √ α β + √ α β + 2 α β α β − α β ⊗ = (cid:48) ⊗ (cid:48) = ˆ3 ⊗ ˆ3 = ˆ3 (cid:48) ⊗ ˆ3 (cid:48) = ⊕ ⊕ ⊕ (cid:48) , ⊗ (cid:48) = ˆ3 ⊗ ˆ3 (cid:48) = (cid:48) ⊕ ⊕ ⊕ (cid:48) , ⊗ ˆ3 = (cid:48) ⊗ ˆ3 (cid:48) = ˆ1 ⊕ ˆ2 ⊕ ˆ3 ⊕ ˆ3 (cid:48) ⊗ ˆ3 (cid:48) = (cid:48) ⊗ ˆ3 = ˆ1 (cid:48) ⊕ ˆ2 ⊕ ˆ3 ⊕ ˆ3 (cid:48) , ˆ1 ∼ α β + α β + α β , ˆ2 ∼ (cid:18) √ α β + √ α β − α β − α β √ α β − √ α β + 2 α β + 2 α β (cid:19) , ˆ3 ∼ α β − α β α β − α β α β − α β (cid:48) , ˆ3 (cid:48) ∼ √ α β + √ α β + α β + α β √ α β − √ α β + α β + α β α β + α β − α β (cid:48) , ˆ1 (cid:48) ∼ α β + α β + α β , ˆ2 ∼ (cid:18) √ α β − √ α β + 2 α β + 2 α β −√ α β − √ α β + 2 α β + 2 α β ) (cid:19) , ˆ3 ∼ √ α β + √ α β + α β + α β √ α β − √ α β + α β + α β α β + α β − α β (cid:48) , ˆ3 (cid:48) ∼ α β − α β α β − α β α β − α β Table 3:
The Kronecker products and Clebsch-Gordan coefficients of the S × Z group. The weight four Siegel modular forms can be constructed from the tensor product of Y ( τ )and Y (cid:48) ( τ ). Using the Clebsch-Gordan coefficients listed in table 3, we find : (cid:40) Y (4) a = Y Y = Y ,Y (4) b = ( Y (cid:48) Y (cid:48) ) = Y + Y + Y , : Y (4) = ( Y (cid:48) Y (cid:48) ) = (cid:18) √ Y Y − Y Y √ Y − Y ) + 4 Y Y (cid:19) , : Y (4) = ( Y (cid:48) Y (cid:48) ) = (0 , , T , (cid:48) : Y (4) (cid:48) a = Y Y (cid:48) = Y ( Y , Y , Y ) T ,Y (4) (cid:48) b = ( Y (cid:48) Y (cid:48) ) (cid:48) = √ Y Y + 2 Y Y √ Y − Y ) + 2 Y Y Y + Y − Y , (B.4)30here Y (4) (cid:48) a and Y (4) (cid:48) b denote the two independent weight 4 modular forms in the representation (cid:48) . Acknowledgements
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