Generalised CP Symmetry in Modular-Invariant Models of Flavour

The formalism of combined finite modular and generalised CP (gCP) symmetries for theories of flavour is developed. The corresponding consistency conditions for the two symmetry transformations acting on the modulus $\tau$ and on the matter fields are derived. The implications of gCP symmetry in theories of flavour based on modular invariance described by finite modular groups are illustrated with the example of a modular $S_4$ model of lepton flavour. Due to the addition of the gCP symmetry, viable modular models turn out to be more constrained, with the modulus $\tau$ being the only source of CP violation.


Introduction
Explaining the flavour structures of quarks and leptons remains to be one of the fundamental problems in particle physics. We still do not know the absolute values of neutrino masses, as well as whether CP symmetry is violated in the lepton sector. However, the observed pattern of neutrino mixing with two large and one small (but non-zero) angles suggests that a non-Abelian discrete flavour symmetry can be at work (see [1][2][3][4] for reviews).
In the bottom-up discrete symmetry approach to lepton flavour, some of the neutrino mixing angles and the Dirac CP violation (CPV) phase δ are generically predicted to be correlated with each other, since all of them are expressed in terms of few free parameters. At the same time, the Majorana phases [5], present in the neutrino mixing matrix if neutrinos are Majorana particles, remain unconstrained. In order to reduce the number of free parameters, the discrete flavour symmetry can be combined with the so-called generalised CP (gCP) symmetry [6,7]. Such models have more predictive power and allow, in particular, for prediction of the Majorana phases. The implications of combining the gCP symmetry with a flavour symmetry have been extensively studied for many discrete groups, including A 4 [7,8], T [9], S 4 [6,[10][11][12][13][14][15] and A 5 [16][17][18][19] (see also [20]).
The conventional bottom-up discrete symmetry approach to lepton flavour has certain drawbacks. Within this approach specific models need to be constructed to obtain predictions for neutrino masses. A flavour symmetry in these models is typically spontaneously broken by vacuum expectation values (VEVs) of scalar flavon fields. Usually, a relatively large number of these fields with a rather complicated potential possessing additional shaping symmetries to achieve the correct vacuum alignment is needed. Possible higher-dimensional operators may affect model predictions, and thus, have to be taken into account.
In view of that, a new approach, in which modular invariance plays the role of flavour symmetry, has been put forward in Ref. [21]. The main feature of this approach is that the Yukawa couplings and fermion mass matrices in the Lagrangian of the theory arise from modular forms which depend on the value of a single complex scalar field τ , called the modulus. In addition, both the couplings and fields transform under a finite modular group Γ N . Once τ acquires a VEV, the Yukawa couplings and the form of the mass matrices get fixed, and a certain flavour structure arises. For N ≤ 5, the finite modular groups are isomorphic to permutation groups (see, e.g., [22]) used to build models of lepton and quark flavours. Until now models based on the finite modular groups Γ 2 S 3 [23,24], Γ 3 A 4 [21,[23][24][25][26][27][28][29], Γ 4 S 4 [30,31] and Γ 5 A 5 [32,33] have been constructed in the literature. In a top-down approach, the interplay of flavour and modular symmetries has recently been considered in the context of string theory in Refs. [34][35][36].
In the present work, we study the implications of combining the gCP symmetry with modular invariance in the construction of models of flavour. It is expected that combining the two symmetries in a model of flavour will lead to a reduction of the number of free parameters, and thus to an increased predictive power of the model. The article is organised as follows. In Section 2, we summarise key features of combining gCP symmetry with a discrete non-Abelian group and briefly describe modular symmetry approach to flavour. Then, in Section 3, requiring consistency between CP and modular symmetries, we derive the action of CP on i) the modulus, ii) superfields and iii) multiplets of modular forms. After that, in Section 4, we discuss implications for the charged lepton and neutrino mass matrices, and determine the values of the modulus which allow for CP conservation. In Section 5, we give an example of a viable model invariant under both the modular and CP symmetries. Finally, we conclude in Section 6.

Generalised CP Symmetry Combined with a Flavour Symmetry
Consider a supersymmetric (SUSY) theory 1 with a flavour symmetry described by a non-Abelian discrete group G f . A chiral superfield ψ(x) in a generic irreducible representation (irrep) r of G f transforms under the action of G f as where ρ r (g) is the unitary representation matrix for the element g in the irrep r. A theory which is also invariant under CP symmetry has to remain unchanged under the following transformation: with a bar denoting the Hermitian conjugate superfield, and where x = (t, x), x P = (t, −x) and X r is a unitary matrix acting on flavour space [37]. The transformation in eq. (2.2) is commonly referred to as a gCP transformation. In the case of X r = 1 r , one recovers the canonical CP transformation. The action of the gCP transformation on a chiral superfield and, in particular, on its fermionic component is described in detail in Appendix A. The form of the matrix X r is constrained due to the presence of a flavour symmetry [6,7]. Performing first a gCP transformation, followed by a flavour symmetry transformation g ∈ G f , and subsequently an inverse gCP transformation, one finds 3) The theory should remain invariant under this sequence of transformations, and thus, the resulting transformation must correspond to a flavour symmetry transformation (cf. eq. (2.1)) ρ r (g ), with g being some element of G f , i.e., we have: X r ρ * r (g)X −1 r = ρ r (g ) , g, g ∈ G f . (2.4) This equation defines the consistency condition, which has to be respected for consistent implementation of a gCP symmetry along with a flavour symmetry, provided the full flavour symmetry group G f has been correctly identified [6,7]. Notice that X r is a unitary matrix defined for each irrep [38]. Several well-known facts about this consistency condition are in order.
• Equation (2.4) has to be satisfied for all irreps r simultaneously, i.e., the elements g and g must be the same for all r.
• For a given irrep r, the consistency condition defines X r up to an overall phase and a G f transformation.
• It follows from eq. (2.4) that the elements g and g must be of the same order.
• It is sufficient to impose eq. (2.4) on the generators of a discrete group G f .
• The chain CP → g → CP −1 maps the group element g onto g and preserves the flavour symmetry group structure. Therefore, it realises a homomorphism v(g) = g of G f . Assuming the presence of faithful representations r, i.e., those for which ρ r maps each element of G f to a distinct matrix, eq. (2.4) defines a unique mapping of G f to itself. In this case, v(g) is an automorphism of G f .
• The automorphism v(g) = g must be class-inverting with respect to G f , i.e. g and g −1 belong to the same conjugacy class [38]. It is furthermore an outer automorphism, meaning no h ∈ G f exists such that g = h −1 gh.
It has been shown in Ref. [6] that under the assumption of X r being a symmetric matrix, 2 the full symmetry group is isomorphic to a semi-direct product G f H CP , where H CP Z CP 2 is the group generated by the gCP transformation.
Finally, we would like to note that for G f = A 4 , S 4 and A 5 and in the bases for their representation matrices summarised in Appendix B.2, the gCP transformation X r = 1 r up to inner automorphisms, i.e, X r = ρ r (g), g ∈ G f , as shown in Refs. [8], [7] and [16], respectively.

Modular Symmetry and Modular-Invariant Theories
In this subsection, we briefly summarise the modular invariance approach to flavour [21]. An element γ of the modular group Γ acts on a complex variable τ belonging to the upper-half complex plane as follows: The modular group Γ is isomorphic to the projective special linear group P SL(2, Z) = SL(2, Z)/Z 2 , where SL(2, Z) is the special linear group of integer 2 × 2 matrices with unit determinant and Z 2 = {I, −I} is its centre, I being the identity element. The group Γ can be presented in terms of two generators S and T satisfying The generators admit the following matrix representation: The action of S and T on τ amounts to inversion with a change of sign and translation, respectively: Let us consider the infinite normal subgroups Γ(N ), N = 1, 2, 3, . . . , of SL(2, Z) (called also the principal congruence subgroups): For N = 1 and 2, one defines the groups Γ(N ) ≡ Γ(N )/{I, −I} (note that Γ(1) ≡ Γ), while for N > 2, Γ(N ) ≡ Γ(N ). The quotient groups Γ N ≡ Γ/Γ(N ) turn out to be finite. They are referred to as finite modular groups. Remarkably, for N ≤ 5, these groups are isomorphic to permutation groups: Γ 2 S 3 , Γ 3 A 4 , Γ 4 S 4 and Γ 5 A 5 . Their group theory is summarised in Appendix B. We recall here only that the group Γ N is presented by two generators S and T satisfying: (2.10) We will work in the basis in which the generators S and T of these groups are represented by symmetric matrices, for all irreducible representations r. The convenience of this choice will become clear later on. For the groups Γ N with N ≤ 5, the working bases are provided in Appendix B.2.
The key elements of the considered framework are modular forms f (τ ) of weight k and level N . These are holomorphic functions, which transform under Γ(N ) as follows: where the weight k is an even and non-negative number, and level N is a natural number. For a certain k and N , the modular forms span a linear space of finite dimension. One can find a basis in this space such that a multiplet of modular forms F (τ ) ≡ (f 1 (τ ), f 2 (τ ), . . . ) T transforms according to a unitary representation r of Γ N : In Appendix C.1, we provide the multiplets of modular forms of lowest non-trivial weight k = 2 at levels N = 2, 3, 4 and 5, i.e., for S 3 , A 4 , S 4 and A 5 . Multiplets of higher weight modular forms can be constructed from tensor products of the lowest weight multiplets. For N = 4 (i.e., S 4 ), we present in Appendix C.3 modular multiplets of weight k ≤ 10 derived in the symmetric basis for the S 4 generators (see Appendix B.2). For N = 3 and N = 5 (i.e., A 4 and A 5 ), modular multiplets of weight up to 6 and 10, computed in the bases employed by us, can be found in [21] and [32], respectively. In the case of N = 1 rigid SUSY, the matter action S reads where K is the Kähler potential, W is the superpotential, ψ denotes a set of chiral supermultiplets ψ i , and τ is the modulus chiral superfield, whose lowest component is the complex scalar field acquiring a VEV. 3 The θ and θ are Graßmann variables. The modulus τ and supermultiplets ψ i transform under the action of the modular group in a certain way [39,40].
Assuming, in addition, that the supermultiplets ψ i = ψ i (x) transform in a certain irreducible representation r i of Γ N , the transformations read: (2.15) 3 We will use the same notation τ for the lowest complex scalar component of the modulus superfield and will call this component also "modulus" since in what follows we will be principally concerned with this scalar field.
It is worth noting that ψ i is not a multiplet of modular forms, and hence, the weight (−k i ) can be odd and/or negative. Invariance of the matter action under these transformations implies where the second line represents a Kähler transformation.
An example Kähler potential, which we will use in what follows, reads: with Λ 0 having mass dimension one. The superpotential can be expanded in powers of ψ i as follows: where 1 stands for an invariant singlet of Γ N . For each set of n fields {ψ i 1 , . . . , ψ in }, the index s labels the independent singlets. Each of these is accompanied by a coupling constant g i 1 ... in,s and is obtained using a modular multiplet Y i 1 ... in,s of the requisite weight. Indeed, to ensure invariance of W under the transformations in eq. (2.15), the set Y i 1 ... in,s (τ ) of functions must transform in the following way (we omit indices for brevity): where r Y is a representation of Γ N , and k Y and r Y are such that Thus, Y i 1 ... in,s (τ ) represents a multiplet of weight k Y and level N modular forms transforming in the representation r Y of Γ N (cf. eq. (2.13)).

gCP Transformations Consistent with Modular Symmetry
As we saw in subsection 2.1, CP transformations can in general be combined with flavour symmetries in a non-trivial way. In the set-up of subsection 2.2, the role of flavour symmetry is played by modular symmetry. In this section, we derive the most general form of a CP transformation consistent with modular symmetry. Unlike the case of discrete flavour symmetries, field transformation properties under CP are restricted to a unique possibility, given the transformation of the modulus (see subsection 3.1) and eq. (2.2). The derivation we are going to present is agnostic to the UV completion of the theory and, in particular, the origin of modular symmetry.

CP Transformation of the Modulus τ
Let us first apply the consistency condition chain 4 to an arbitrary chiral superfield ψ(x) assigned to an irreducible unitary representation r of Γ N , which transforms as ψ(x) → X r ψ(x P ) under CP: where τ CP −1 is the result of applying CP −1 to the modulus τ . The resulting transformation should be equivalent to a modular transformation γ which depends on γ and maps ψ(x) to (c τ + d ) −k ρ r (γ )ψ(x). Taking this into account, we get Since the matrices X r , ρ r (γ) and ρ r (γ ) are independent of τ , the overall coefficient on the right-hand side has to be a constant: 5 where λ ∈ C, and |λ| = 1 due to unitarity of ρ r (γ) and ρ r (γ ). The values of λ, c and d depend on γ.
Taking γ = S, so that c = 1, d = 0, and denoting c (S) = C, d (S) = D while keeping henceforth the notation λ(S) = λ, we find τ = (λτ * CP −1 − D)/C, and consequently, Let us now act with the chain CP → T → CP −1 on the modulus τ itself: The resulting transformation has to be a modular transformation, therefore λ/C ∈ Z. Since |λ| = 1, we immediately find |C| = 1, λ = ±1. After choosing the sign of C as C = ∓1 so that Im τ CP > 0, the CP transformation rule (3.5) simplifies to with n ∈ Z. One can easily check that the chain CP → S → CP −1 = γ (S) (applied to the modulus τ itself) imposes no further restrictions on the form of τ CP . Since S and T generate the entire modular group, we conclude that eq. (3.7) is the most general CP transformation of the modulus τ compatible with the modular symmetry. It is always possible to redefine the CP transformation in such a way that n = 0. Consider the composition CP ≡ T −n • CP so that τ CP −−→ −τ * . It is worth noting that this redefinition represents an inner automorphism which does not spoil the form of gCP transformation in eq. (2.2). Indeed, the chiral superfields transform under CP as Thus, CP has the same properties as the original CP transformation up to a redefinition of X r . Therefore, from now on we will assume without loss of generality that the modulus τ transforms under CP as 6 τ It obviously follows from the preceding equation that τ does not change under the action of Thus, in what concerns the action on the modulus τ we have: CP 2 = I, CP −1 = CP .

Extended Modular Group
Having derived the explicit form of the CP transformation for the modulus τ , we are now in a position to find the action of CP on the modular group Γ as an outer automorphism u(γ).
For any modular transformation γ ∈ Γ we have This implies that the sought-after automorphism is In particular, one has (CP ) S (CP ) −1 = S, (CP ) T (CP ) −1 = T −1 , or simply u(S) = S and u(T ) = T −1 . It is straightforward to check that the mapping (3.12) is indeed an outer automorphism of Γ. 7 Notice further that if γ ∈ Γ(N ), then also u(γ) ∈ Γ(N ). By adding the CP transformation (3.9) as a new generator to the modular group, one obtains the so-called extended modular group: (see, e.g., [44]), which has a structure of a semi-direct product Γ * The CP transformation of the modulus derived by us from the requirement of consistency between modular and CP symmetries has appeared in the context of string-inspired models (see, e.g., Refs. [36,[41][42][43]). 7 One can explicitly check that i) u(γ1)u(γ2) = u(γ1γ2), meaning u is an automorphism, and that ii) there is no group element γ ∈ Γ such that u(γ) = γ −1 γ γ, meaning that u is an outer automorphism.
with determinant ±1, the matrices M and −M being identified. The CP transformation is then represented by the matrix 14) The action of Γ * on the complex upper-half plane is defined as

CP Transformation of Chiral Superfields
A chiral superfield ψ(x) transforms according to eq. (2. 2) under CP. The consistency condition chain (3.1) applied to ψ constrains the form of its CP transformation matrix X r as in eq. (3.3), with the overall coefficient on the right-hand side being constant, as discussed earlier, see eq. (3.4). Since λ = ±1, the coefficient on the right-hand side of eq. (3.3) is (±1) k . This sign is actually determined by the signs of matrices in the outer automorphism (3.12), which are unphysical in a modular symmetric theory. By choosing 8 these signs in such a way that c = −c, d = d, in accordance with eq. (3.12), we obtain a trivial coefficient +1, and the constraint on X r reduces to The constraint we get coincides with the corresponding constraint in the case of non-Abelian discrete flavour symmetries, eq. (2.4). However, unlike in the usual discrete flavour symmetry approach, modular symmetry restricts the form of the automorphism γ → γ = u(γ) to the unique possibility given in eq. (3.12), which acts on the generators as S → u(S) = S and T → u(T ) = T −1 . Therefore, for each irreducible representation r, X r in eq. (3.16) is fixed up to an overall phase by Schur's lemma.
For the working bases discussed in subsection 2.2 and given in Appendix B.2, one has X r = 1 r , i.e., the gCP transformation has the canonical form. The key feature of the aforementioned bases which allows for this simplification is that the group generators S and T are represented by symmetric matrices. Indeed, if eq. (2.11) holds, one has: so that X r = 1 r solves the consistency condition (3.16).

CP Transformation of Modular Multiplets
Since modular multiplets Y (τ ) transform under the modular group in essentially the same way as chiral superfields, it is natural to expect that the above discussion holds for modular multiplets as well. In particular, they should transform under CP as Y → X r Y * . Still, it is instructive to derive their transformation rule explicitly.
Under a modular transformation, Y (τ ) transforms as in eq. (2.19), while under the action of CP one has Y (τ ) → Y (−τ * ). It can be shown (see Appendix D of [31]) that the complexconjugated CP-transformed multiplets Y * (−τ * ) transform almost like the original multiplets Y (τ ) under a modular transformation, namely: for a multiplet Y (τ ) of weight k transforming in the irreducible representation r of Γ N . Using the consistency condition in eq. (3.16), one then sees that it is the object X T r Y * (−τ * ) which transforms like Y (τ ) under a modular transformation, i.e.: If there exists a unique modular multiplet at a certain level N , weight k and representation r, then proportionality follows: with z ∈ C. This is indeed the case for 2 ≤ N ≤ 5 and lowest weight k = 2. Since which can be absorbed in the normalisation of Y (τ ), and ii) that X r must be symmetric in this case, X r X * r = 1 r ⇒ X r = X T r , independently of the basis. One can then write for these multiplets, as anticipated.
As we have seen in subsection 3.3, in a basis in which the generators S and T of Γ N are represented by symmetric matrices, one has X r = 1 r . From eq. (3.20) it follows that Y (−τ * ) = e iφ Y * (τ ), the phase φ being removable, as commented above. At the q-expansion level this means that, in such a basis, all the expansion coefficients are real up to a common complex phase. This is indeed the case for the lowest-weight modular multiplets of Γ N with N ≤ 5, as can be explicitly verified from the q-expansions collected in Appendix C.2. This is further the case for the higher-weight modular multiplets of these groups in such a basis, given the reality of Clebsch-Gordan coefficients, summarised in Appendix B.3.

Implications of CP Invariance for the Couplings
We have found so far that a CP transformation consistent with modular symmetry acts on fields and modular multiplets in the following way: A SUSY modular-invariant theory is thus CP-conserving if the transformation (4.1) leaves the matter action S given by eq. (2.14) unchanged. In particular, the superpotential W has to transform into its Hermitian conjugate, while the Kähler potential K is allowed to change by a Kähler transformation.
The Kähler potential of eq. (2.17) is clearly invariant under the CP transformation (4.1), since it depends on |ψ| 2 and Im τ , both of which remain unchanged (up to a change x → x P which does not affect S). On the other hand, the superpotential can be written as a sum of independent terms of the form where Y s (τ ) are modular multiplets of a certain weight and irreducible representation, and g s are complex coupling constants, see eq. (2.18). Such terms transform non-trivially under CP, which leads to a certain constraint on the couplings g s .
This can be easily checked for a symmetric basis, as in this basis X r = 1 r for any representation r, so that one has (assuming proper normalisation of the modular multiplets Y s (τ )) where in the last equality we have used the reality of the Clebsch-Gordan coefficients, which holds for N ≤ 5. It is now clear that a term in the sum of eq. (4.2) transforms into the Hermitian conjugate of which should coincide with the original term. The independence of singlets now implies the constraint g s = g * s , i.e., all coupling constants g s have to be real to conserve CP. Let us denote byg s the couplings written for a general basis and arbitrary normalisation of the modular multiplets. The CP constraint ong s is then more complicated, since the singlets of different bases coincide only up to normalisation factors, determined by the choice of normalisations of the Clebsch-Gordan coefficients and of the modular multiplets. Since the normalisation factors can differ between singlets, the corresponding couplingsg s may require non-trivial phases to conserve CP. These phases can be found directly by performing a basis transformation and matchingg s to g s in the symmetric basis (and with proper modular multiplet normalisation).

Implications of CP Invariance for the Mass Matrices
As a more concrete example, let us consider the Yukawa coupling term which gives rise to the charged lepton mass matrix. Here E c is a modular symmetry multiplet of SU (2) charged lepton singlets, L is a modular symmetry multiplet of SU (2) lepton doublets, and H d is a Higgs doublet which transforms trivially under modular symmetry and whose neutral component acquires a VEV v d = H 0 d after electroweak symmetry breaking. Expanding the singlets, one gets where entries of the matrices λ s ij (τ ) are formed from components of the corresponding modular multiplets Y s (τ ). In a general basis, superfields transform under CP as and we set η d = 1 without loss of generality. It follows that so that CP conservation implies The resulting charged lepton mass matrix which coincides with the corresponding constraint in the case of CP invariance combined with discrete flavour symmetry, apart from the fact that now the mass matrix depends on the modulus τ which also transforms under CP. Similarly, for the neutrino Majorana mass matrix M ν one has Note that matrix X L is the same in eqs. (4.10) and (4.11) since left-handed charged leptons l L and left-handed neutrinos ν lL form an electroweak SU (2) doublet L, so they transform uniformly both under CP and modular transformations: This can also be found directly from the form of the charged current (CC) weak interaction Lagrangian by ensuring its CP invariance. 9 In a symmetric basis X L = X R = 1, the constraints on the mass matrices simplify to which further reduce to reality of the couplings. Namely, for the charged lepton mass matrix one has Clearly, CP invariance requires g s = g * s , since λ s (τ ) are linearly independent matrices, which in turn is guaranteed by independence of the singlets.

CP-Conserving Values of the Modulus τ
In a CP-conserving modular-invariant theory both CP and modular symmetry are broken spontaneously by the VEV of the modulus τ . However, there exist certain values of τ which conserve CP, while breaking the modular symmetry. Obviously, this is the case if τ is left invariant by CP, i.e. τ CP − − → −τ * = τ , (4.16) meaning that τ lies on the imaginary axis, Re τ = 0. In a symmetric basis one then has as can be seen from eq. (4.14). The resulting mass matrices are real and the corresponding CPV phases are trivial, such that sin δ = sin α 21 = sin α 31 = 0 in the standard parametrisation [45] of the PMNS mixing matrix.
Let us now consider a point γτ in the plane of the modulus related to a CP-invariant point τ = −τ * by a modular transformation γ. This point is physically equivalent to τ due to modular invariance and therefore it should also be CP-conserving. However, γτ does not go to itself under CP. Instead, one has for some γ ∈ Γ. 10 Indeed, one can check that modular invariance of the mass terms requires the mass matrices to transform under a modular transformation as where ρ L and ρ E are the representation matrices for the SU (2) lepton doublet L and charged lepton singlets E c , respectively. We have also taken into account the rescaling of fields due to the non-canonical form of the Kähler potential (2.17), which leads to cancellation of the modular weights in the transformed mass matrices. It is clear from eq. (4.20) that mass eigenvalues are unaffected by the replacement τ → γτ in the mass matrices. Moreover, the unitary rotations U e and U ν diagonalising the mass matrices M e M † e and M ν respectively transform as so the PMNS mixing matrix U PMNS = U † e U ν does not change. This means that the mass matrices evaluated at points τ and γτ lead to the same values of physical observables.
If we now consider a value of τ which satisfies eq. from eq. (4.14). It follows from the above discussion that the observables evaluated at τ coincide with their complex conjugates, hence CPV phases are trivial (0 or π).
To find all points satisfying eq. (4. 19), it is sufficient to restrict ourselves to the fundamental domain D of the modular group given by since all other points are physically equivalent to the points from D.
The interior of D, which we denote as int(D), maps to itself under CP. Apart from that, no two points from int(D) are related by any modular transformation. Therefore, if τ ∈ int(D), then eq. (4.19) reduces to eq. (4.16) and we find again Re τ = 0. The remaining possibility is that τ lies on the boundary of D. Then it is easy to show that it also satisfies eq.

Example: CP-Invariant Modular S 4 Models
To illustrate the use of gCP invariance combined with modular symmetry for model building, we consider modular Γ 4 S 4 models of lepton masses and mixing, in which neutrino masses are generated via the type I seesaw mechanism. Such models have been extensively studied in Ref. [31] in the context of plain modular symmetry without gCP invariance. Here we briefly summarise the construction of Ref. [31] and investigate additional constraints on the models imposed by CP invariance, having the modulus τ as the only potential source of CPV.
It has been found in Ref. [31] that the minimal (in terms of the total number of parameters) viable choice of modular weights and representations is which leads to the superpotential of the form where the multiplets of modular forms Y 2,3 and Y 3,3 have been derived in Ref. [30]. Here no sums are implied, since each singlet is unique, and the coefficients (α, β, γ) = (α 1 , α 2 , α 3 ), g and Λ are real without loss of generality, as the corresponding phases can be absorbed into the fields E c 1 , E c 2 , E c 3 , L and N c , respectively. Therefore, the only complex parameter of the theory is g /g. If a symmetric basis is used and the modular multiplets are properly normalised, then CP is conserved whenever Im g /g = 0 . (5.4) The basis used in Ref. [31] is not symmetric. One can check that it can be related to the symmetric basis here considered by the following transformation matrices U r : 3 ) 1 written in different bases, and taking into account an extra factor of i arising from the normalisation of the modular multiplets used in Ref. [31], we find that also in this basis CP invariance results in the condition (5.4). In what follows, we report the parameter values in the basis of Ref. [31] for ease of comparison.
Through numerical search, five viable pairs of regions of the parameter space have been found in Ref. [31], denoted as A and A * , B and B * , etc. with the starred regions corresponding to CP-conjugated models τ → −τ * , (g /g) → (g /g) * predicting the opposite values of the CPV phases. Among these five pairs of regions only one pair (A and A * , for which ρ N ∼ 3 ) is consistent with the condition (5.4), and only a small portion of the parameter space is allowed. We report the corresponding best fit values and the confidence intervals of the parameters and observables in Table 1.
This minimal CP-invariant model, predicting 12 observables, is characterised by 7 parameters: the 6 real parameters v d α, β/α, γ/α, v 2 u g 2 /Λ, g /g, Im τ and the phase Re τ . 11 The three real parameters v d α, β/α and γ/α are fixed by fitting the three charged lepton masses.  The remaining three real parameters v 2 u g 2 /Λ, g /g, Im τ and the phase Re τ describe the nine neutrino observables: three neutrino masses, three neutrino mixing angles and three CPV phases.
As a result, this model has more predictive power than the original model from Ref. [31], which is described by the same parameters and an additional phase arg(g /g). In fact, the correlations between sin 2 θ 23 , the neutrino masses and the CPV phases, which were present in the original model, now reduce to accurate predictions of these observables at a few percent level. This can be seen by comparing the ranges from Table 1 of the present article with  Table 5a and Fig. 2 of Ref. [31]. Apart from that, many correlations between pairs of observables and between observables and parameters arise. We report these correlations in Figs. 1 and 2.
We also check numerically that CP invariance is restored for the CP-conserving values of τ derived in subsection 4.3. To achieve this, we vary the value of τ while keeping all other parameters fixed to their best fit values, and present the resulting sin 2 δ(τ ), sin 2 α 21 (τ ) and sin 2 α 31 (τ ) as heatmap plots in the τ plane in Fig. 3. Notice that this variation is done for illustrative purposes only, as it spoils the values of the remaining observables. Those are in agreement with experimental data only in a small region of the τ plane [31]. The sine-squared of a phase measures the strength of CPV, with the value of 0 (shown with green colour) corresponding to no CPV and the value of 1 (shown with red colour) corresponding to maximal CPV. As anticipated, both the boundary of D and the imaginary axis conserve CP, appearing in green colour in Fig. 3. However, even a small departure from a CP-conserving value of τ can lead to large CPV due to strong dependence of the observables on τ . This is noticeably the case in the vicinity of the boundary of the fundamental domain.

Summary and Conclusions
In the present article we have developed the formalism of combining modular and generalised CP (gCP) symmetries for theories of flavour. To this end the corresponding consistency conditions for the two symmetry transformations acting on the modulus τ and on the matter fields were derived. We have shown that these consistency conditions imply that under the CP transformation the modulus τ → −τ * , without loss of generality. This transformation extends the modular group Γ P SL(2, Z) to Γ * Γ Z CP 2 P GL(2, Z). Considering the cases of the finite modular groups Γ N with N = 2, 3, 4, 5, which are isomorphic to the non-Abelian discrete groups S 3 , A 4 , S 4 and A 5 , respectively, we have demonstrated that the gCP transformation matrix X r realising a rotation in flavour space when acting on a multiplet ψ(x) as ψ(x) → X r ψ(x P ), where x P = (t, −x), can always be chosen to be the identity matrix 1 r . Assuming this choice and a proper normalisation of multiplets of modular forms Y (τ ), transforming in irreducible representations of the groups Γ N with N = 2, 3, 4, 5, we have shown that under CP these multiplets get complex conjugated. As a consequence, we have found that gCP invariance implies that the constants g, which accompany each invariant singlet in the superpotential, must be real. Thus, the number of free parameters in modular-invariant models which also enjoy a gCP symmetry gets reduced, leading to a higher predictive power of such models. In these models, the only source of both modular symmetry breaking and CP violation is the VEV of the modulus τ . We have further demonstrated that CP is conserved for the values of the modulus at the boundary of the fundamental domain and on the imaginary axis.
Finally, via the example of a modular S 4 model of lepton flavour with type I seesaw mechanism of neutrino mass generation, we have illustrated the results obtained in the present study regarding the implications of gCP symmetry in modular-invariant theories of flavour. This model was considered in Ref. [31] without the requirement of gCP invariance. We have shown that imposing the latter leads to much reduced ranges of allowed values of the neutrino mass and mixing parameters as well as to much stronger correlations between the different neutrino-related observables (Table 1 and Figs. 1 and 2).

A Spinors and Superfields under CP
For clarity, in this Appendix alone chiral superfields will be denoted with a tilde to distinguish them from scalar and fermion fields. Under CP, a Dirac 4-spinor flavour multiplet Ψ transforms as −x), X Ψ is a unitary matrix in flavour space and C is the charge conjugation matrix, satisfying C −1 γ µ C = − γ T µ and C T = −C. We can further consider C to be unitary, C −1 = C † , without loss of generality. The factor of i in eq. (A.1) is a convention employed consistently throughout this Appendix. Spacetime coordinates are henceforth omitted. For the two-component formalism widely used in SUSY we are going to discuss in the present Appendix it proves convenient to consider the Weyl basis for γ µ matrices. In this basis, as in the usual Dirac basis, the matrix C is real, so that C = −C T = −C † = −C −1 .
One may write the Dirac spinor Ψ in terms of two generally independent Weyl 2-spinors, with dotted and undotted indices shown explicitly. Bars on 2-spinors denote conjugation, e.g. ψα = δα α (ψ α ) * . Notice that for 4-spinors Ψ ≡ Ψ † A, with A numerically equal to γ 0 but with a different index structure. One has: Numerically C = iγ 0 γ 2 , while γ 5 ≡ iγ 0 γ 1 γ 2 γ 3 = diag(−1, 1). The chiral projection operators are defined as P L,R = (1 ∓ γ 5 )/2. From the CP transformation (A.1) of a Dirac spinor Ψ, one can obtain the CP transformation of the Weyl spinors ψ and φ in eq. (A.2): For the (chiral) fields in the lepton sector, in particular, one has: where iL , e iR and ν iR (L i , E c i and N c i ) denote lepton doublet, charged lepton singlet and neutrino singlet 4(2)-spinors, respectively. It is then straightforward to find the transformation of a pair of contracted spinors, e.g.: (A.10) In the framework of rigid N = 1 SUSY in 4 dimensions, taking the Graßmann coordinates θ α and θα to transform under CP as all other Weyl spinors, i.e.
one can obtain a consistent CP transformation of a chiral superfield ψ = ϕ + √ 2 θψ − θ 2 F , with F being an auxiliary field and ϕ denoting the scalar lowest component of the superfield, expected to transform under CP in the same way as the latter. Indeed, using eqs. (A.11) and (A.5), one sees that consistency implies and where the bar denotes the Hermitian conjugated superfield. We thus see that the superpotential, up to unitary rotations in flavour space, is exchanged under CP with its conjugate. 12 Given the CP transformations of spinors in eqs. (A.7) -(A.9), one then has, for the chiral superfields L i , E c i and N c i in the lepton sector: 6  12  24  60  irreps 1 , 1 , 2 1, 1 , 1 , 3 1, 1 , 2, 3, 3 1, 3, 3 , 4, 5 B Group Theory of Γ N ≤5

B.1 Order and Irreducible Representations
The finite modular groups Γ N = Γ/Γ(N ) P SL(2, Z N ), with N > 1, can be defined by two generators S and T satisfying the relations: The order of these groups is given by (see, e.g., [46]): where the product is over prime divisors p of N . It is straightforward to show that |Γ N | is even for all N . Decomposing N > 2 in its unique prime factorisation, N = k i=1 p n i i with n i ≥ 1, one has: The group order will be even if 2|Γ N >2 | is a multiple of 4. This is trivially verified when N is a power of 2. If N is not a power of 2, then at least one of its prime factors is odd, say p j . Since (n 2 − 1) ≡ 0 (mod 4) for odd n, it follows that the group order is even also in this case, as (p 2 j − 1) divides 2|Γ N >2 |. In what follows, we focus on the case N ≤ 5. The orders and irreducible representations of these groups are listed in Table 2.

B.3 Clebsch-Gordan Coefficients
For completeness, and for each level N = 2, 3, 4, 5, we also reproduce here the nontrivial Clebsch-Gordan coefficients in the symmetric basis of Appendix B.2. Entries of each multiplet entering the tensor product are denoted by α i and β i .
Below, for each N ≤ 5, we reproduce explicit expressions for the lowest weight modular multiplets Y (N,k=2) r as vectors of the aforementioned log-derivatives. Note that these multiplets are given in the symmetric basis of Appendix B.2. For these groups, the dimensions of linear spaces at different weights and the irreducible representations arising at lowest weight are summarised in Table 3.