Generalised CP Symmetry in Modular-Invariant Models of Flavour
SSISSA 14/2019/FISIIPMU19-0080IPPP/19/46CFTP/19-019
Generalised CP Symmetry inModular-Invariant Models of Flavour
P. P. Novichkov a, , J. T. Penedo b, , S. T. Petcov a,c, , A. V. Titov d, a SISSA/INFN, Via Bonomea 265, 34136 Trieste, Italy b CFTP, Departamento de F´ısica, Instituto Superior T´ecnico, Universidade de Lisboa,Avenida Rovisco Pais 1, 1049-001 Lisboa, Portugal c Kavli IPMU (WPI), University of Tokyo, 5-1-5 Kashiwanoha, 277-8583 Kashiwa, Japan d Institute for Particle Physics Phenomenology, Department of Physics, Durham University,South Road, Durham DH1 3LE, United Kingdom
Abstract
The formalism of combined finite modular and generalised CP (gCP) symmetries fortheories of flavour is developed. The corresponding consistency conditions for the twosymmetry transformations acting on the modulus τ and on the matter fields are derived.The implications of gCP symmetry in theories of flavour based on modular invariancedescribed by finite modular groups are illustrated with the example of a modular S modelof lepton flavour. Due to the addition of the gCP symmetry, viable modular models turnout to be more constrained, with the modulus τ being the only source of CP violation. E-mail: [email protected] E-mail: [email protected] Also at Institute of Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences, 1784 Sofia,Bulgaria. E-mail: [email protected] a r X i v : . [ h e p - ph ] A ug Introduction
Explaining the flavour structures of quarks and leptons remains to be one of the fundamentalproblems 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 observedpattern of neutrino mixing with two large and one small (but non-zero) angles suggests thata non-Abelian discrete flavour symmetry can be at work (see [1–4] for reviews).In the bottom-up discrete symmetry approach to lepton flavour, some of the neutrinomixing angles and the Dirac CP violation (CPV) phase δ are generically predicted to becorrelated 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 neu-trinos are Majorana particles, remain unconstrained. In order to reduce the number of freeparameters, 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, forprediction of the Majorana phases. The implications of combining the gCP symmetry with aflavour symmetry have been extensively studied for many discrete groups, including A [7, 8], T (cid:48) [9], S [6, 10–15] and A [16–19] (see also [20]).The conventional bottom-up discrete symmetry approach to lepton flavour has certaindrawbacks. Within this approach specific models need to be constructed to obtain predictionsfor neutrino masses. A flavour symmetry in these models is typically spontaneously broken byvacuum expectation values (VEVs) of scalar flavon fields. Usually, a relatively large numberof these fields with a rather complicated potential possessing additional shaping symmetriesto achieve the correct vacuum alignment is needed. Possible higher-dimensional operatorsmay 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 flavoursymmetry, has been put forward in Ref. [21]. The main feature of this approach is thatthe Yukawa couplings and fermion mass matrices in the Lagrangian of the theory arise frommodular 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, anda certain flavour structure arises. For N ≤
5, the finite modular groups are isomorphicto 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 (cid:39) S [23, 24], Γ (cid:39) A [21, 23–29],Γ (cid:39) S [30, 31] and Γ (cid:39) A [32, 33] have been constructed in the literature. In a top-downapproach, the interplay of flavour and modular symmetries has recently been considered inthe context of string theory in Refs. [34–36].In the present work, we study the implications of combining the gCP symmetry withmodular invariance in the construction of models of flavour. It is expected that combiningthe two symmetries in a model of flavour will lead to a reduction of the number of freeparameters, and thus to an increased predictive power of the model. The article is organisedas follows. In Section 2, we summarise key features of combining gCP symmetry with adiscrete non-Abelian group and briefly describe the modular symmetry approach to flavour.Then, in Section 3, requiring consistency between CP and modular symmetries, we derive theaction of CP on i) the modulus, ii) superfields and iii) multiplets of modular forms. Afterthat, 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, wegive an example of a viable model invariant under both the modular and CP symmetries.1inally, we conclude in Section 6. Consider a supersymmetric (SUSY) theory 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 ψ ( x ) g −→ ρ r ( g ) ψ ( x ) , g ∈ G f , (2.1)where ρ r ( g ) is the unitary representation matrix for the element g in the irrep r . A theorywhich is also invariant under CP symmetry has to remain unchanged under the followingtransformation: ψ ( x ) CP −−→ X r ψ ( x P ) , (2.2)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) iscommonly referred to as a gCP transformation. In the case of X r = r , one recovers thecanonical CP transformation. The action of the gCP transformation on a chiral superfieldand, 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 ψ ( x ) CP −−→ X r ψ ( x P ) g −→ X r ρ ∗ r ( g ) ψ ( x P ) CP − −−−−→ X r ρ ∗ r ( g ) X − r ψ ( x ) . (2.3)The theory should remain invariant under this sequence of transformations, and thus, theresulting transformation must correspond to a flavour symmetry transformation (cf. eq. (2.1)) ρ r ( g (cid:48) ), with g (cid:48) being some element of G f , i.e., we have: X r ρ ∗ r ( g ) X − r = ρ r ( g (cid:48) ) , g, g (cid:48) ∈ G f . (2.4)This equation defines the consistency condition, which has to be respected for consistentimplementation of a gCP symmetry along with a flavour symmetry, provided the full flavoursymmetry group G f has been correctly identified [6, 7]. Notice that X r is a unitary matrixdefined for each irrep [38]. Several well-known facts about this consistency condition are inorder. • Equation (2.4) has to be satisfied for all irreps r simultaneously, i.e., the elements g and g (cid:48) 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 (cid:48) must be of the same order. In what follows we will focus on a supersymmetric construction with global supersymmetry (see subsec-tion 2.2). It is sufficient to impose eq. (2.4) on the generators of a discrete group G f . • The chain CP → g → CP − maps the group element g onto g (cid:48) and preserves theflavour symmetry group structure. Therefore, it realises a homomorphism v ( g ) = g (cid:48) of G f . Assuming the presence of faithful representations r , i.e., those for which ρ r mapseach element of G f to a distinct matrix, eq. (2.4) defines a unique mapping of G f toitself. In this case, v ( g ) is an automorphism of G f . • The automorphism v ( g ) = g (cid:48) must be class-inverting with respect to G f , i.e. g (cid:48) and g − belong to the same conjugacy class [38]. It is furthermore an outer automorphism,meaning no h ∈ G f exists such that g (cid:48) = h − gh .It has been shown in Ref. [6] that under the assumption of X r being a symmetric matrix, the full symmetry group is isomorphic to a semi-direct product G f (cid:111) H CP , where H CP (cid:39) Z CP is the group generated by the gCP transformation.Finally, we would like to note that for G f = S , A , S and A and in the bases for theirrepresentation matrices summarised in Appendix B.2, the gCP transformation X r = r upto inner automorphisms, i.e, X r = ρ r ( g ) , g ∈ G f , as shown in Refs. [8], [7] and [16]. In this subsection, we briefly summarise the modular invariance approach to flavour [21]. Anelement γ of the modular group Γ acts on a complex variable τ belonging to the upper-halfcomplex plane as follows: γτ = aτ + bcτ + d , where a, b, c, d ∈ Z and ad − bc = 1 , Im τ > . (2.5)The modular group Γ is isomorphic to the projective special linear group P SL (2 , Z ) = SL (2 , Z ) / Z , where SL (2 , Z ) is the special linear group of integer 2 × Z = { I, − I } is its centre, I being the identity element. The group Γ canbe presented in terms of two generators S and T satisfying S = ( ST ) = I . (2.6)The generators admit the following matrix representation: S = (cid:18) −
11 0 (cid:19) , T = (cid:18) (cid:19) . (2.7)The action of S and T on τ amounts to inversion with a change of sign and translation,respectively: τ S −→ − τ , τ T −→ τ + 1 . (2.8)Let us consider the infinite normal subgroups Γ( N ), N = 1 , , , . . . , of SL (2 , Z ) (calledalso the principal congruence subgroups):Γ( N ) = (cid:26)(cid:18) a bc d (cid:19) ∈ SL (2 , Z ) , (cid:18) a bc d (cid:19) = (cid:18) (cid:19) (mod N ) (cid:27) . (2.9) In this case, the CP transformation applied twice to a chiral superfield gives the field itself (see eq. (2.2)).We note also that the CP transformation applied twice to a fermion field gives the field multiplied by a minussign, which is consistent with the superfield transformation (see Appendix A). N = 1 and 2, one defines the groups Γ( N ) ≡ Γ( N ) / { I, − I } (note that Γ(1) ≡ Γ), whilefor
N >
2, Γ( N ) ≡ Γ( N ). The quotient groups Γ N ≡ Γ / Γ( N ) turn out to be finite. Theyare referred to as finite modular groups. Remarkably, for N ≤
5, these groups are isomorphicto permutation groups: Γ (cid:39) S , Γ (cid:39) A , Γ (cid:39) S and Γ (cid:39) A . Their group theoryis summarised in Appendix B. We recall here only that the group Γ N is presented by twogenerators S and T satisfying: S = ( ST ) = T N = I . (2.10)We will work in the basis in which the generators S and T of these groups are represented bysymmetric matrices, ρ r ( S ) = ρ T r ( S ) , ρ r ( T ) = ρ T r ( T ) , (2.11)for all irreducible representations r . The convenience of this choice will become clear lateron. 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 andlevel N . These are holomorphic functions, which transform under Γ( N ) as follows: f ( γτ ) = ( cτ + d ) k f ( τ ) , γ ∈ Γ( N ) , (2.12)where the weight k is an even and non-negative number, and the level N is a natural number.For certain k and N , the modular forms span a linear space of finite dimension. One canfind a basis in this space such that a multiplet of modular forms F ( τ ) ≡ ( f ( τ ) , f ( τ ) , . . . ) T transforms according to a unitary representation r of Γ N : F ( γτ ) = ( cτ + d ) k ρ r ( γ ) F ( τ ) , γ ∈ Γ . (2.13)In Appendix C.1, we provide the multiplets of modular forms of lowest non-trivial weight k = 2 at levels N = 2 , , S , A , S and A . Multiplets of higher weightmodular forms can be constructed from tensor products of the lowest weight multiplets. For N = 4 (i.e., S ), we present in Appendix C.3 modular multiplets of weight k ≤
10 derived inthe symmetric basis for the S generators (see Appendix B.2). For N = 3 and N = 5 (i.e., A and A ), modular multiplets of weight up to 6 and 10, computed in the bases employedby us, can be found in [21] and [32], respectively.In the case of N = 1 rigid SUSY, the matter action S reads S = (cid:90) d x d θ d θ K ( τ, τ , ψ, ψ ) + (cid:90) d x d θ W ( τ, ψ ) + (cid:90) d x d θ W ( τ , ψ ) , (2.14)where K is the K¨ahler potential, W is the superpotential, ψ denotes a set of chiral supermulti-plets ψ i , and τ is the modulus chiral superfield, whose lowest component is the complex scalarfield acquiring a VEV. 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, inaddition, that the supermultiplets ψ i = ψ i ( x ) transform in a certain irreducible representation r i of Γ N , the transformations read: γ = (cid:18) a bc d (cid:19) ∈ Γ : τ → aτ + bcτ + d ,ψ i → ( cτ + d ) − k i ρ r i ( γ ) ψ i . (2.15) We will use the same notation τ for the lowest complex scalar component of the modulus superfield andwill call this component also “modulus” since in what follows we will be principally concerned with this scalarfield.
4t is worth noting that ψ i is not a multiplet of modular forms, and hence, the weight ( − k i )can be positive or negative, and can be even or odd. Invariance of the matter action underthese transformations implies W ( τ, ψ ) → W ( τ, ψ ) ,K ( τ, τ , ψ, ψ ) → K ( τ, τ , ψ, ψ ) + f K ( τ, ψ ) + f K ( τ , ψ ) , (2.16)where the second line represents a K¨ahler transformation.An example of the K¨ahler potential, which we will use in what follows, reads: K ( τ, τ , ψ, ψ ) = − Λ log( − iτ + iτ ) + (cid:88) i | ψ i | ( − iτ + iτ ) k i , (2.17)with Λ having mass dimension one. The superpotential can be expanded in powers of ψ i asfollows: W ( τ, ψ ) = (cid:88) n (cid:88) { i ,...,i n } (cid:88) s g i ... i n ,s ( Y i ... i n ,s ( τ ) ψ i . . . ψ i n ) ,s , (2.18)where stands for an invariant singlet of Γ N . For each set of n fields { ψ i , . . . , ψ i n } , theindex s labels the independent singlets. Each of these is accompanied by a coupling constant g i ... i n ,s and is obtained using a modular multiplet Y i ... i n ,s of the requisite weight. Indeed,to ensure invariance of W under the transformations in eq. (2.15), the set Y i ... i n ,s ( τ ) offunctions must transform in the following way (we omit indices for brevity): Y ( τ ) γ −→ ( cτ + d ) k Y ρ r Y ( γ ) Y ( τ ) , (2.19)where r Y is a representation of Γ N , and k Y and r Y are such that k Y = k i + · · · + k i n , (2.20) r Y ⊗ r i ⊗ . . . ⊗ r i n ⊃ . (2.21)Thus, Y i ... i n ,s ( τ ) represents a multiplet of weight k Y and level N modular forms transformingin the representation r Y of Γ N (cf. eq. (2.13)). As we saw in subsection 2.1, CP transformations can in general be combined with flavoursymmetries in a non-trivial way. In the set-up of subsection 2.2, the role of flavour symmetryis played by modular symmetry. In this section, we derive the most general form of a CPtransformation consistent with modular symmetry. Unlike the case of discrete flavour sym-metries, field transformation properties under CP are restricted to a unique possibility, giventhe transformation of the modulus (see subsection 3.1) and eq. (2.2). The derivation we aregoing to present is agnostic to the UV completion of the theory and, in particular, the originof modular symmetry. 5 .1 CP Transformation of the Modulus τ Let us first apply the consistency condition chain CP → γ → CP − = γ (cid:48) ∈ Γ (3.1)to an arbitrary chiral superfield ψ ( x ) assigned to an irreducible unitary representation r ofΓ N , which transforms as ψ ( x ) → X r ψ ( x P ) under CP: ψ ( x ) CP −−→ X r ψ ( x P ) γ −→ ( cτ ∗ + d ) − k X r ρ ∗ r ( γ ) ψ ( x P ) CP − −−−−→ ( cτ ∗ CP − + d ) − k X r ρ ∗ r ( γ ) X − r ψ ( x ) , (3.2)where τ CP − is the result of applying CP − to the modulus τ . The resulting transformationshould be equivalent to a modular transformation γ (cid:48) which depends on γ and maps ψ ( x ) to( c (cid:48) τ + d (cid:48) ) − k ρ r ( γ (cid:48) ) ψ ( x ). Taking this into account, we get X r ρ ∗ r ( γ ) X − r = (cid:18) c (cid:48) τ + d (cid:48) cτ ∗ CP − + d (cid:19) − k ρ r ( γ (cid:48) ) . (3.3)Since the matrices X r , ρ r ( γ ) and ρ r ( γ (cid:48) ) are independent of τ , the overall coefficient on theright-hand side has to be a constant: c (cid:48) τ + d (cid:48) cτ ∗ CP − + d = 1 λ ∗ , (3.4)where λ ∈ C , and | λ | = 1 due to unitarity of ρ r ( γ ) and ρ r ( γ (cid:48) ). The values of λ , c (cid:48) and d (cid:48) depend on γ .Taking γ = S , so that c = 1, d = 0, and denoting c (cid:48) ( S ) = C , d (cid:48) ( S ) = D while keepinghenceforth the notation λ ( S ) = λ , we find τ = ( λτ ∗ CP − − D ) /C , and consequently, τ CP − −−−−→ τ CP − = λ ( Cτ ∗ + D ) , τ CP −−→ τ CP = 1 C ( λτ ∗ − D ) . (3.5)Let us now act with the chain CP → T → CP − on the modulus τ itself: τ CP −−→ C ( λτ ∗ − D ) T −→ C ( λ ( τ ∗ + 1) − D ) CP − −−−−→ τ + λC . (3.6)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 = ∓ τ CP >
0, the CP transformation rule (3.5) simplifies to τ CP −−→ n − τ ∗ , (3.7)with n ∈ Z . One can easily check that the chain CP → S → CP − = γ (cid:48) ( S ) (applied to themodulus τ itself) imposes no further restrictions on the form of τ CP . Since S and T generate It may be possible to generalise the CP transformation such that it can be combined not only withmodular but also with other internal symmetries of the theory. We are not going to consider this case here. Strictly speaking, this is only true for non-zero weights k . We assume that at least one superfield withnon-zero modular weight exists in the theory, because otherwise the modulus has no effect on the superfieldtransformations, and the modular symmetry approach reduces to the well-known discrete symmetry approachto flavour. τ compatible with the modular symmetry.It is always possible to redefine the CP transformation in such a way that n = 0. Considerthe composition CP (cid:48) ≡ T − n ◦ CP so that τ CP (cid:48) −−→ − τ ∗ . It is worth noting that this redefinitionrepresents an inner automorphism which does not spoil the form of gCP transformation ineq. (2.2). Indeed, the chiral superfields transform under CP (cid:48) as ψ CP (cid:48) −−→ ρ − n r ( T ) X r ψ . (3.8)Thus, CP (cid:48) 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 τ CP −−→ − τ ∗ . (3.9)It obviously follows from the preceding equation that τ does not change under the action of CP : τ CP −−−→ τ . (3.10)Thus, in what concerns the action on the modulus τ we have: CP = I , CP − = CP . Having derived the explicit form of the CP transformation for the modulus τ , we are now ina 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 τ CP −−→ − τ ∗ γ −→ − aτ ∗ + bcτ ∗ + d CP − −−−−→ aτ − b − cτ + d . (3.11)This implies that the sought-after automorphism is γ = (cid:18) a bc d (cid:19) → u ( γ ) ≡ CP γ CP − = (cid:18) a − b − c d (cid:19) . (3.12)In particular, one has ( CP ) S ( CP ) − = S , ( CP ) T ( CP ) − = T − , or simply u ( S ) = S and u ( T ) = T − . It is straightforward to check that the mapping (3.12) is indeed an outerautomorphism of Γ. 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, oneobtains the so-called extended modular group :Γ ∗ = (cid:68) τ T −→ τ + 1 , τ S −→ − /τ, τ CP −−→ − τ ∗ (cid:69) (3.13)(see, e.g., [44]), which has a structure of a semi-direct product Γ ∗ (cid:39) Γ (cid:111) Z CP , with Z CP = { I, τ → − τ ∗ } . The group Γ ∗ is isomorphic to the group P GL (2 , Z ) of integral 2 × The CP transformation of the modulus derived by us from the requirement of consistency between modularand CP symmetries has appeared in the context of string-inspired models (see, e.g., Refs. [36, 41–43]). One can explicitly check that i) u ( γ ) u ( γ ) = u ( γ γ ), meaning u is an automorphism, and that ii) thereis no group element (cid:101) γ ∈ Γ such that u ( γ ) = (cid:101) γ − γ (cid:101) γ , meaning that u is an outer automorphism. ±
1, the matrices M and − M being identified. The CP transformation isthen represented by the matrix CP = (cid:18) − (cid:19) , CP γ CP − = u ( γ ) . (3.14)The action of Γ ∗ on the complex upper-half plane is defined as (cid:18) a bc d (cid:19) ∈ Γ ∗ : τ → aτ + bcτ + d if ad − bc = 1 ,τ → aτ ∗ + bcτ ∗ + d if ad − bc = − . (3.15) A chiral superfield ψ ( x ) transforms according to eq. (2.2) under CP. The consistency conditionchain (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, seeeq. (3.4). Since λ = ±
1, the coefficient on the right-hand side of eq. (3.3) is ( ± k . Thissign is actually determined by the signs of matrices in the outer automorphism (3.12), whichare unphysical in a modular symmetric theory. By choosing these signs in such a way that c (cid:48) = − c , d (cid:48) = d , in accordance with eq. (3.12), we obtain a trivial coefficient +1, and theconstraint on X r reduces to X r ρ ∗ r ( γ ) X − r = ρ r ( γ (cid:48) ) . (3.16)The constraint we get coincides with the corresponding constraint in the case of non-Abeliandiscrete flavour symmetries, eq. (2.4). However, unlike in the usual discrete flavour symmetryapproach, modular symmetry restricts the form of the automorphism γ → γ (cid:48) = u ( γ ) to theunique possibility given in eq. (3.12), which acts on the generators as S → u ( S ) = S and T → u ( T ) = T − . Therefore, for each irreducible representation r , X r in eq. (3.16) is fixedup 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 = r , i.e., the gCP transformation has the canonical form. The key feature of the aforementionedbases which allows for this simplification is that the group generators S and T are representedby symmetric matrices. Indeed, if eq. (2.11) holds, one has: ρ ∗ r ( S ) = ρ † r ( S ) = ρ r ( S − ) = ρ r ( S ) , ρ ∗ r ( T ) = ρ † r ( T ) = ρ r ( T − ) , (3.17)so that X r = r solves the consistency condition (3.16). Since modular multiplets Y ( τ ) transform under the modular group in essentially the sameway as chiral superfields, it is natural to expect that the above discussion holds for modularmultiplets as well. In particular, they should transform under CP as Y → X r Y ∗ . Still, it isinstructive to derive their transformation rule explicitly. This choice is possible since in the presence of fields with an odd modular weight the theory automaticallyrespects a sign-flip Z symmetry, which acts only non-trivially on these fields. This implies that the discussedsign is unphysical. Y ( τ ) transforms as in eq. (2.19), while under the actionof CP one has Y ( τ ) → Y ( − τ ∗ ). It can be shown (see Appendix D of [31]) that the complex-conjugated CP-transformed multiplets Y ∗ ( − τ ∗ ) transform almost like the original multiplets Y ( τ ) under a modular transformation, namely: Y ∗ ( − τ ∗ ) γ −→ Y ∗ ( − ( γτ ) ∗ ) = Y ∗ ( u ( γ )( − τ ∗ )) = ( cτ + d ) k ρ ∗ r ( u ( γ )) Y ∗ ( − τ ∗ ) , (3.18)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.: X T r Y ∗ ( − τ ∗ ) γ −→ ( cτ + d ) k ρ r ( γ ) (cid:2) X T r Y ∗ ( − τ ∗ ) (cid:3) . (3.19)If there exists a unique modular form multiplet at a certain level N , weight k and repre-sentation r , then proportionality follows: Y ( τ ) = z X T r Y ∗ ( − τ ∗ ) , (3.20)with z ∈ C . This is indeed the case for 2 ≤ N ≤ k = 2. Since Y ( − ( − τ ∗ ) ∗ ) = Y ( τ ), it follows that X r X ∗ r = | z | r , implying i) that z = e iφ is a phasewhich can be absorbed in the normalisation of Y ( τ ), and ii) that X r must be symmetric inthis case, X r X ∗ r = r ⇒ X r = X T r , independently of the basis. One can then write Y ( τ ) CP −−→ Y ( − τ ∗ ) = X r Y ∗ ( τ ) (3.21)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 = 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 -expansionlevel this means that, in such a basis, all the expansion coefficients are real up to a commoncomplex phase. This is indeed the case for the lowest-weight modular form 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 abasis, given the reality of Clebsch-Gordan coefficients, summarised in Appendix B.3. We have found so far that a CP transformation consistent with modular symmetry acts onfields and modular form multiplets in the following way: τ CP −−→ − τ ∗ , ψ ( x ) CP −−→ X r ψ ( x P ) , Y ( τ ) CP −−→ Y ( − τ ∗ ) = X r Y ∗ ( τ ) . (4.1)A SUSY modular-invariant theory is thus CP-conserving if the transformation (4.1) leavesthe matter action S given by eq. (2.14) unchanged. In particular, the superpotential W hasto transform into its Hermitian conjugate, while the K¨ahler potential K is allowed to changeby a K¨ahler transformation. 9he K¨ahler potential of eq. (2.17) is clearly invariant under the CP transformation (4.1),since it depends on | ψ | 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 ofindependent terms of the form W ⊃ (cid:88) s g s ( Y s ( τ ) ψ . . . ψ n ) ,s , (4.2)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 = r for any repre-sentation r , so that one has (assuming proper normalisation of the modular multiplets Y s ( τ )) g s ( Y s ( τ ) ψ . . . ψ n ) ,s CP −−→ g s (cid:0) Y s ( − τ ∗ ) ψ . . . ψ n (cid:1) ,s = g s (cid:0) Y ∗ s ( τ ) ψ . . . ψ n (cid:1) ,s = g s ( Y s ( τ ) ψ . . . ψ n ) ,s , (4.3)where in the last equality we have used the reality of the Clebsch-Gordan coefficients, whichholds for N ≤
5. It is now clear that a term in the sum of eq. (4.2) transforms into theHermitian conjugate of g ∗ s ( Y s ( τ ) ψ . . . ψ n ) ,s , (4.4)which should coincide with the original term due to the independence of singlets in eq. (4.2).It now follows that g s = g ∗ s , i.e., all coupling constants g s have to be real to conserve CP.As a final remark, let us denote by ˜ g s the couplings written for a general basis andarbitrary normalisation of the modular form multiplets. The CP constraint on ˜ g s is thenmore complicated, since the singlets of different bases coincide only up to normalisationfactors, determined by the choice of normalisations of the Clebsch-Gordan coefficients andof the modular form multiplets. Since the normalisation factors can differ between singlets,the corresponding couplings ˜ g s may require non-trivial phases to conserve CP. These phasescan be found directly by performing a basis transformation and matching ˜ g s to g s in thesymmetric basis (and with proper modular form multiplet normalisation). As a more concrete example, let us consider the Yukawa coupling term W L = (cid:88) s g s ( Y s ( τ ) E c LH d ) ,s , (4.5)which gives rise to the charged lepton mass matrix. Here E c is a modular symmetry multipletof 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 whoseneutral component acquires a VEV v d = (cid:104) H d (cid:105) after electroweak symmetry breaking.Expanding the singlets, one gets W L = (cid:88) s g s λ sij ( τ ) E ci L j H d ≡ λ ij ( τ ) E ci L j H d , (4.6)10here entries of the matrices λ sij ( τ ) are formed from components of the corresponding modularmultiplets Y s ( τ ). In a general basis, superfields transform under CP as E c CP −−→ X ∗ R E c , L CP −−→ X L L , H d CP −−→ η d H d , (4.7)and we set η d = 1 without loss of generality. It follows that W L CP −−→ (cid:16) X † R λ ( − τ ∗ ) X L (cid:17) ij E ci L j H d , (4.8)so that CP conservation implies X † R λ ( − τ ∗ ) X L = λ ∗ ( τ ) . (4.9)The resulting charged lepton mass matrix M e = v d λ † (written in the left-right convention)satisfies X † L M e ( − τ ∗ ) X R = M ∗ e ( τ ) , (4.10)which coincides with the corresponding constraint in the case of CP invariance combinedwith discrete flavour symmetry, apart from the fact that now the mass matrix depends onthe modulus τ which also transforms under CP. Similarly, for the neutrino Majorana massmatrix M ν one has X TL M ν ( − τ ∗ ) X L = M ∗ ν ( τ ) . (4.11)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 transformuniformly both under CP and modular transformations: X l L = X ν lL ≡ X L , ρ l L ( γ ) = ρ ν lL ( γ ) ≡ ρ L ( γ ) , k l L = k ν lL ≡ k L . (4.12)This can also be found directly from the form of the charged current (CC) weak interactionLagrangian L CC = − g √ (cid:88) l = e,µ,τ l L γ α ν lL W α † + h.c. (4.13)by ensuring its CP invariance. In a symmetric basis X L = X R = , the constraints on the mass matrices simplify to M e ( − τ ∗ ) = M ∗ e ( τ ) , M ν ( − τ ∗ ) = M ∗ ν ( τ ) , (4.14)which further reduce to reality of the couplings. Namely, for the charged lepton mass matrixone has M e ( − τ ∗ ) = v d (cid:88) s g ∗ s ( λ s ) † ( − τ ∗ ) = v d (cid:88) s g ∗ s ( λ s ) T ( τ ) ,M ∗ e ( τ ) = (cid:32) v d (cid:88) s g ∗ s ( λ s ) † ( τ ) (cid:33) ∗ = v d (cid:88) s g s ( λ s ) T ( τ ) . (4.15)Clearly, CP invariance requires g s = g ∗ s , since λ s ( τ ) are linearly independent matrices, whichin turn is guaranteed by independence of the singlets. Since the original superfields in a modular-invariant theory are not normalised canonically, there is actuallya prefactor of (2 Im τ ) − k L in the CC weak interaction Lagrangian which originates from the K¨ahler potential.This prefactor is necessary for modular invariance in order to compensate the weights k l L = k ν lL ≡ k L . .3 CP-Conserving Values of the Modulus τ In a CP-conserving modular-invariant theory both CP and modular symmetry are brokenspontaneously by the VEV of the modulus τ . However, there exist certain values of τ whichconserve CP, while breaking the modular symmetry. Obviously, this is the case if τ is leftinvariant 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 M e ( τ ) = M ∗ e ( τ ) , M ν ( τ ) = M ∗ ν ( τ ) , (4.17)as can be seen from eq. (4.14). The resulting mass matrices are real and the correspondingCPV phases are trivial, such that sin δ = sin α = sin α = 0 in the standard parametrisa-tion [45] of the PMNS mixing matrix.Let us now consider a point γτ in the plane of the modulus related to a CP-invariantpoint τ = − τ ∗ by a modular transformation γ . This point is physically equivalent to τ dueto modular invariance and therefore it should also be CP-conserving. However, γτ does notgo to itself under CP. Instead, one has γτ CP −−→ ( γτ ) CP = u ( γ ) τ CP = u ( γ ) τ = u ( γ ) γ − γτ , (4.18)so the resulting CP-transformed value ( γτ ) CP is related to the original value γτ by a modulartransformation u ( γ ) γ − .Hence, it is natural to expect that a value of τ conserves CP if it is left invariant by CP up to a modular transformation , i.e., τ CP −−→ − τ ∗ = γτ (4.19)for some γ ∈ Γ. Indeed, one can check that modular invariance of the mass terms requiresthe mass matrices to transform under a modular transformation as M e ( τ ) γ −→ M e ( γτ ) = ρ L ( γ ) M e ( τ ) ρ TE ( γ ) ,M ν ( τ ) γ −→ M ν ( γτ ) = ρ ∗ L ( γ ) M ν ( τ ) ρ † L ( γ ) , (4.20)where ρ L and ρ E are the representation matrices for the SU (2) lepton doublet L and chargedlepton singlets E c , respectively. We have also taken into account the rescaling of fields dueto the non-canonical form of the K¨ahler potential (2.17), which leads to cancellation of themodular 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 massmatrices M e M † e and M ν respectively transform as U e γ −→ ρ L U e , U ν γ −→ ρ L U ν , (4.21)so the PMNS mixing matrix U PMNS = U † e U ν does not change. This means that the massmatrices evaluated at points τ and γτ lead to the same values of physical observables. A similar condition has been derived in Ref. [42] in the context of string theories (in which the CPsymmetry represents a discrete gauge symmetry), postulating the action of CP on the compactified directions.
12f we now consider a value of τ which satisfies eq. (4.19), then in a symmetric basis wehave M e ( γτ ) = M ∗ e ( τ ) , M ν ( γτ ) = M ∗ ν ( τ ) , (4.22)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 funda-mental domain D of the modular group given by D = (cid:26) τ ∈ C : Im τ > , | Re τ | ≤ , | τ | ≥ (cid:27) , (4.23)since all other points are physically equivalent to the points from D . All CP-conserving valuesof τ outside the fundamental domain are related to the CP-conserving values of τ within thefundamental domain by modular transformations.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 non-trivial modular transformation. Therefore,if τ ∈ int( D ), then eq. (4.19) reduces to eq. (4.16) and we find again Re τ = 0. The remainingpossibility is that τ lies on the boundary of D . Then it is easy to show that it also satisfieseq. (4.19), but with a non-trivial γ . Namely, for the left vertical line we have τ = − / i y CP −−→ / i y = T τ , while for the arc we have τ = e iϕ CP −−→ − e − iϕ = Sτ .We conclude that CP-conserving values of τ are the imaginary axis and the boundaryof the fundamental domain D . These values can also be obtained by algebraically solvingeq. (4.19), by noticing that it implies u ( γ ) γ = I and using the requirement that γτ belongsto the fundamental domain.Thus, at the left and right vertical boundaries as well as at the arc boundary of thefundamental domain and at the imaginary axis within it, i.e., at τ LB = − / i y , τ RB =1 / i y , y ≥ √ /
2, at τ AB = e iϕ , ϕ = [60 ◦ , ◦ ], and at τ IA = i y , y ≥
1, we have a Z CP symmetry. At the left and right cusps of the fundamental domain, τ L = − / i √ / τ R = 1 / i √ /
2, and at τ C = i , the symmetry is enhanced to Z CP × Z ST , Z CP × Z T S and Z CP × Z S , respectively [31], while at τ T = i ∞ , the symmetry in the case of Γ N is enhancedto Z CP × Z TN [29, 31]. S Models
To illustrate the use of gCP invariance combined with modular symmetry for model building,we consider modular Γ (cid:39) S models of lepton masses and mixing, in which neutrino massesare generated via the type I seesaw mechanism. Such models have been extensively studied inRef. [31] in the context of plain modular symmetry without gCP invariance. Here we brieflysummarise the construction of Ref. [31] and investigate additional constraints on the modelsimposed by CP invariance, having the modulus τ as the only potential source of CPV.Representations of the superfields under the modular group are chosen as follows: • Higgs doublets H u and H d (with (cid:104) H u (cid:105) = v u ) are S trivial singlets of zero weight: ρ u = ρ d ∼ and k u = k d = 0; • lepton SU (2) doublets L and neutral lepton gauge singlets N c form S triplets ( ρ L ∼ or (cid:48) , ρ N ∼ or (cid:48) ) of weights k L and k N , respectively;13 charged lepton singlets E c , E c and E c transform as S singlets ( ρ , , ∼ or (cid:48) ) ofweights k , , .The relevant superpotential terms read W = (cid:88) i =1 α i (cid:16) E ci L Y ( k αi ) (cid:17) H d + g (cid:16) N c L Y ( k g ) (cid:17) H u + Λ (cid:16) N c N c Y ( k Λ ) (cid:17) , (5.1)where Y ( k ) denotes a modular multiplet of weight k and level 4, and a sum over all independentsinglets with the coefficients α i = ( α i , α (cid:48) i , . . . ), g = ( g, g (cid:48) , . . . ) and Λ = (Λ , Λ (cid:48) , . . . ) is implied.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 k α = 2 , k α = k α = 4 , k g = 2 , k Λ = 0 ,ρ L ∼ , ρ ∼ (cid:48) , ρ ∼ , ρ ∼ (cid:48) , ρ N ∼ or (cid:48) , (5.2)which leads to the superpotential of the form W = α (cid:16) E c L Y (2) (cid:48) (cid:17) H d + β (cid:16) E c L Y (4) (cid:17) H d + γ (cid:16) E c L Y (4) (cid:48) (cid:17) H d + g (cid:16) N c L Y (2) (cid:17) H u + g (cid:48) (cid:16) N c L Y (2) (cid:48) (cid:17) H u + Λ ( N c N c ) , (5.3)where the multiplets of modular forms Y (2) , (cid:48) and Y (4) , (cid:48) have been derived in Ref. [30]. Here nosums are implied, since each singlet is unique, and the coefficients ( α, β, γ ) = ( α , α , α ), g and Λ are real without loss of generality, as the corresponding phases can be absorbed intothe fields E c , E c , E c , L and N c , respectively. Therefore, the only complex parameter of thetheory is g (cid:48) /g . If a symmetric basis is used and the modular form multiplets are properlynormalised, then CP is conserved wheneverIm (cid:0) g (cid:48) /g (cid:1) = 0 . (5.4)The basis used in Ref. [31] is not symmetric. One can check that it can be related to thesymmetric basis here considered by the following transformation matrices U r : U = U (cid:48) = 1 , U = 1 √ (cid:18) − i i (cid:19) ,U = U (cid:48) = 12 √ − e − iπ/
00 0 − e iπ/ − √ − √ − − √ √ . (5.5)By direct comparison of the singlets ( N c L Y (2) ) and ( N c L Y (2) (cid:48) ) written in different bases,and taking into account an extra factor of i arising from the normalisation of the modularform multiplets used in Ref. [31], we find that also in this basis CP invariance results in thecondition (5.4). In what follows, we report the parameter values in the basis of Ref. [31] forease of comparison.Through numerical search, five viable pairs of regions of the parameter space have beenfound in Ref. [31], denoted as A and A ∗ , B and B ∗ , etc. with the starred regions correspondingto CP-conjugated models τ → − τ ∗ , ( g (cid:48) /g ) → ( g (cid:48) /g ) ∗ predicting the opposite values of the14est fit value 2 σ range 3 σ rangeRe τ ± . ± (0 . − . ± (0 . − . τ . − .
017 1 . − . β/α . − .
02 7 . − . γ/α . − . . − . g (cid:48) /g − . − (0 . − . − (0 . − . v d α [MeV] 53.61 v u g / Λ [eV] 0.0135 m e /m µ . − . . − . m µ /m τ . − . . − . r . − . . − . δm [10 − eV ] 7.326 7 . − .
551 6 . − . | ∆ m | [10 − eV ] 2.457 2 . − .
489 2 . − . θ . − . . − . θ . − . . − . θ . − . . − . m [eV] 0.01211 0 . − . . − . m [eV] 0.01483 0 . − . . − . m [eV] 0.05139 0 . − . . − . (cid:80) i m i [eV] 0.07833 0 . − . . − . |(cid:104) m (cid:105)| [eV] 0.01201 0 . − . . − . δ/π ± . ± (1 . − . ± (1 . − . α /π ± . ± (0 . − . ± (0 . − . α /π ± . ± (1 . − . ± (1 . − . N σ
Table 1:
Best fit values along with 2 σ and 3 σ ranges of the parameters and observables inthe minimal CP-invariant modular S model. CP symmetry is spontaneously broken by theVEV of the modulus τ . .25 0.30 0.35 sin θ s i n θ sin θ ± δ / π sin θ ± α / π sin θ ± α / π sin θ ± α / π sin θ ± α / π sin θ ± δ / π sin θ ± α / π sin θ ± α / π sin θ | › m fi | , e V ± δ/π ± α / π ± δ/π ± α / π Figure 1:
Correlations between pairs of observables in the minimal CP-invariant modular S model. .62 1.64 1.66 ± δ/π | › m fi | , e V ± α /π ± α / π r Σ m , e V r | › m fi | , e V Σ m , eV0.01160.01180.01200.0122 | › m fi | , e V τ s i n θ τ r τ Σ m , e V g /g s i n θ g /g s i n θ g /g δ / π Nσ Figure 2:
Correlations between pairs of observables (continued from Fig. 1) and betweenobservables and parameters in the minimal CP-invariant modular S model. ∗ , for which ρ N ∼ (cid:48) ) isconsistent 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 parametersand observables in Table 1.This minimal CP-invariant model, predicting 12 observables, is characterised by 7 param-eters: the 6 real parameters v d α , β/α , γ/α , v u g / Λ, g (cid:48) /g , Im τ and the phase Re τ . Thethree 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 u g / Λ, g (cid:48) /g , Im τ and the phase Re τ describe the nineneutrino observables: three neutrino masses, three neutrino mixing angles and three CPVphases.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 (cid:48) /g ). In fact, thecorrelations between sin θ , the neutrino masses and the CPV phases, which were present inthe original model, now reduce to accurate predictions of these observables at a few percentlevel. This can be seen by comparing the ranges from Table 1 of the present article withTable 5a and Fig. 2 of Ref. [31]. Apart from that, many correlations between pairs of observ-ables and between observables and parameters arise. We report these correlations in Figs. 1and 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 otherparameters fixed to their best fit values, and present the resulting sin δ ( τ ), sin α ( τ ) andsin α ( τ ) as heatmap plots in the τ plane in Fig. 3. Notice that this variation is donefor illustrative purposes only, as it spoils the values of the remaining observables. Thoseare in agreement with experimental data only in a small region of the τ plane [31]. Thesine-squared of a phase measures the strength of CPV, with the value of 0 (shown with greencolour) corresponding to no CPV and the value of 1 (shown with red colour) corresponding tomaximal 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-conservingvalue of τ can lead to large CPV due to strong dependence of the observables on τ . This isnoticeably the case in the vicinity of the boundary of the fundamental domain. In the present article we have developed the formalism of combining modular and generalisedCP (gCP) symmetries for theories of flavour. To this end the corresponding consistencyconditions for the two symmetry transformations acting on the modulus τ and on the matterfields were derived. We have shown that these consistency conditions imply that under theCP transformation the modulus τ → n − τ ∗ with integer n , and one can choose n = 0without loss of generality. This transformation extends the modular group Γ (cid:39) P SL (2 , Z )to Γ ∗ (cid:39) Γ (cid:111) Z CP (cid:39) P GL (2 , Z ). Considering the cases of the finite modular groups Γ N with N = 2 , , ,
5, which are isomorphic to the non-Abelian discrete groups S , A , S and A , respectively, we have demonstrated that the gCP transformation matrix X r realisinga 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 r . Assuming this choice Re τ should be treated as a phase since the dependence of Yukawa couplings and fermion mass matriceson τ arises through powers of exp(2 πiτ /N ) = exp( πiτ / .5 0.0 0.5Re τ I m τ sin δ τ I m τ sin α τ I m τ sin α Figure 3:
CP violation strength, measured as sine-squared of the CPV phases, for differentvalues of the modulus τ . The boundary of the fundamental domain D defined in eq. (4.23) andthe imaginary axis Re τ = 0 conserve CP. The ranges of Re τ and Im τ are chosen to extendslightly beyond the fundamental domain D to make its boundary clearly visible. and a proper normalisation of multiplets of modular forms Y ( τ ), transforming in irreduciblerepresentations of the groups Γ N with N = 2 , , ,
5, we have shown that under CP thesemultiplets get complex conjugated. As a consequence, we have found that gCP invarianceimplies 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 alsoenjoy a gCP symmetry gets reduced, leading to a higher predictive power of such models. Inthese models, the only source of both modular symmetry breaking and CP violation is theVEV of the modulus τ . We have further demonstrated that CP is conserved for the valuesof the modulus at the boundary of the fundamental domain and on the imaginary axis.Finally, via the example of a modular S model of lepton flavour with type I seesawmechanism of neutrino mass generation, we have illustrated the results obtained in the presentstudy 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 haveshown that imposing the latter leads to much reduced ranges of allowed values of the neutrinomass and mixing parameters as well as to much stronger correlations between the differentneutrino-related observables (Table 1 and Figs. 1 and 2). Acknowledgements
P.P.N. would like to thank Kavli IPMU, where part of this work was carried out, for itshospitality. This project has received funding from the European Union’s Horizon 2020 re-search and innovation programme under the Marie Sk(cid:32)lodowska-Curie grant agreements No674896 (ITN Elusives) and No 690575 (RISE InvisiblesPlus). This work was supported inpart by the INFN program on Theoretical Astroparticle Physics (P.P.N. and S.T.P.) andby the World Premier International Research Center Initiative (WPI Initiative, MEXT),Japan (S.T.P.). The work of J.T.P. was supported by Funda¸c˜ao para a Ciˆencia e a Tecnolo-gia (FCT, Portugal) through the projects CFTP-FCT Unit 777 (UID/FIS/00777/2013 andUID/FIS/00777/2019) and PTDC/FIS-PAR/29436/2017 which are partially funded throughPOCTI (FEDER), COMPETE, QREN and EU.19
Spinors and Superfields under CP
For clarity, in this Appendix alone chiral superfields will be denoted with a tilde to distinguishthem from scalar and fermion fields. Under CP, a 4-component spin-1 / i ( x ) CP −−→ i ( X Ψ ) ij γ C Ψ jT ( x P ) , (A.1)where x = ( t, x ), x P = ( t, − x ), X Ψ is a unitary matrix in flavour space and C is the chargeconjugation matrix, satisfying C − γ µ C = − γ Tµ , which implies C T = − C . These two proper-ties of C do not depend on the representation of the γ -matrices. We can further consider C to be unitary, C − = C † , without loss of generality. The factor of i in eq. (A.1) is a conven-tion employed consistently throughout this Appendix. Spacetime coordinates are henceforthomitted. For the two-component formalism widely used in SUSY we are going to discuss inthe present Appendix it proves convenient to consider the Weyl basis for γ µ matrices. In thisbasis, the matrix C is real, so that C = − C † = − C − .One may write a 4-component spinor Ψ in terms of two Weyl 2-spinors,Ψ = (cid:32) ψ α φ ˙ α (cid:33) , Ψ c ≡ C Ψ T = (cid:32) φ α ψ ˙ α (cid:33) , (A.2)with dotted and undotted indices shown explicitly. Bars on 2-spinors denote conjugation,e.g. ψ ˙ α = δ ˙ αα ( ψ α ) ∗ . Notice that for 4-spinors Ψ ≡ Ψ † A , with A numerically equal to γ butwith a different index structure. One has: A = (cid:18) δ α ˙ β δ ˙ αβ (cid:19) , C = (cid:18) (cid:15) αβ (cid:15) ˙ α ˙ β (cid:19) , with (cid:15) ˙ α ˙ β = − (cid:15) αβ = (cid:18) − (cid:19) , (A.3) γ µ = (cid:18) σ µ ) α ˙ β ( σ µ ) ˙ αβ (cid:19) ⇒ γ = (cid:32) σ ) α ˙ β ( σ ) ˙ αβ (cid:33) . (A.4)Disregarding the spinor-index structure, one has C = iγ γ , γ ≡ iγ γ γ γ = diag( − , ), σ µ = ( σ , σ i ) = ( σ , − σ i ) and σ µ = ( σ , σ i ), with σ = and the σ i being the Pauli matrices.The chiral projection operators are defined as P L,R = ( ∓ γ ) / ψ and φ in eq. (A.2): ψ iα CP −−→ i ( X Ψ ) ij ( σ ) α ˙ β ψ ˙ βj ⇒ ψ ˙ αi CP −−→ i ( X Ψ ) ∗ ij ( σ ) ˙ αβ ψ jβ , (A.5) φ ˙ αi CP −−→ i ( X Ψ ) ij ( σ ) ˙ αβ φ jβ ⇒ φ iα CP −−→ i ( X Ψ ) ∗ ij ( σ ) α ˙ β φ ˙ βj . (A.6)For the (chiral) fields in the lepton sector, in particular, one has: (cid:96) iL ≡ (cid:18) L iα (cid:19) CP −−→ i ( X L ) ij γ C (cid:96) jLT ⇒ L iα CP −−→ i ( X L ) ij ( σ ) α ˙ β L j ˙ β , (A.7) e iR ≡ (cid:32) E ci ˙ α (cid:33) CP −−→ i ( X R ) ij γ C e jRT ⇒ E ci α CP −−→ i ( X R ) ∗ ij ( σ ) α ˙ β E cj ˙ β , (A.8) The matrix C = iγ γ has the same form in the usual Dirac representation of γ -matrices. iR ≡ (cid:32) N ci ˙ α (cid:33) CP −−→ i ( X N ) ij γ C ν jRT ⇒ N ci α CP −−→ i ( X N ) ∗ ij ( σ ) α ˙ β N cj ˙ β , (A.9)where (cid:96) iL , e iR and ν iR ( L i , E ci and N ci ) denote lepton doublet, charged lepton singlet and neu-trino singlet 4(2)-spinors, respectively. It is then straightforward to find the transformationof a pair of contracted spinors, e.g.: E ci L j = (cid:15) αβ E ciβ L jα CP −−→ ( X R ) ∗ ik ( X L ) jl (cid:104) − ( σ T ) ˙ βα (cid:15) αβ ( σ ) β ˙ α (cid:105) E ck ˙ α L l ˙ β = ( X R ) ∗ ik ( X L ) jl E ck L l . (A.10)In the framework of rigid N = 1 SUSY in 4 dimensions, taking the Graßmann coordinates θ α and θ ˙ α to transform under CP as all other Weyl spinors, i.e. θ α CP −−→ i ( σ ) α ˙ β θ ˙ β ⇒ θ ˙ α CP −−→ i ( σ ) ˙ αβ θ β , (A.11)one can obtain a consistent CP transformation of a chiral superfield (cid:101) ψ = ϕ + √ θψ − θ 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 (cid:101) ψ i CP −−→ ( X Ψ ) ij (cid:101) ψ j , (A.12)with ϕ i CP −−→ ( X Ψ ) ij ϕ ∗ j and F i CP −−→ ( X Ψ ) ij F ∗ j , since θ CP −−→ θ , and where the bar denotes theHermitian conjugated superfield. We thus see that the superpotential, up to unitary rotationsin flavour space, is exchanged under CP with its conjugate. Given the CP transformations of spinors in eqs. (A.7) – (A.9), one then has, for the chiralsuperfields (cid:101) L i , (cid:101) E ci and (cid:101) N ci in the lepton sector: (cid:101) L i CP −−→ ( X L ) ij (cid:101) L j ⇒ (cid:101) L i CP −−→ ( X L ) ∗ ij (cid:101) L j , (A.13) (cid:101) E ci CP −−→ ( X R ) ∗ ij (cid:101) E cj ⇒ (cid:101) E ci CP −−→ ( X R ) ij (cid:101) E cj , (A.14) (cid:101) N ci CP −−→ ( X N ) ∗ ij (cid:101) N cj ⇒ (cid:101) N ci CP −−→ ( X N ) ij (cid:101) N cj . (A.15) One can check that the Graßmann integration (cid:82) d θ is exchanged with its conjugate (cid:82) d θ . N Γ (cid:39) S Γ (cid:39) A Γ (cid:39) S Γ (cid:39) A | Γ N | , (cid:48) , , (cid:48) , (cid:48)(cid:48) , , (cid:48) , , , (cid:48) , , (cid:48) , , Table 2:
Orders and irreducible representations for finite modular groups Γ N with N ≤ B Group Theory of Γ N ≤ B.1 Order and Irreducible Representations
The finite modular groups Γ N = Γ / Γ( N ) (cid:39) P SL (2 , Z N ), with N >
1, can be defined by twogenerators S and T satisfying the relations: S = T N = ( ST ) = I . (B.1)The order of these groups is given by (see, e.g., [46]): | Γ N | = N = 2 ,N (cid:89) p | N (cid:18) − p (cid:19) for N > , (B.2)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 > N = (cid:81) ki =1 p n i i with n i ≥
1, one has:2 | Γ N> | = N k (cid:89) i =1 p n i − i (cid:0) p i − (cid:1) . (B.3)The group order will be even if 2 | Γ N> | 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 − ≡ n , it follows that the group order is even also in this case,as ( p j −
1) divides 2 | Γ N> | .In what follows, we focus on the case N ≤
5. The orders and irreducible representationsof these groups are listed in Table 2.
B.2 Symmetric Basis for Group Generators
For the groups S , A , S and A , we explicitly give below the basis for representation matricesin which the group generators S and T are represented by symmetric matrices, see eq. (2.11),for all irreducible representations r (see, e.g., Refs. [2, 21, 47, 48]). B.2.1 Γ (cid:39) S : ρ ( S ) = 1 , ρ ( T ) = 1 , (B.4) (cid:48) : ρ ( S ) = − , ρ ( T ) = − , (B.5) : ρ ( S ) = 12 (cid:18) − −√ −√ (cid:19) , ρ ( T ) = (cid:18) − (cid:19) . (B.6)22 .2.2 Γ (cid:39) A : ρ ( S ) = 1 , ρ ( T ) = 1 , (B.7) (cid:48) : ρ ( S ) = 1 , ρ ( T ) = ω , (B.8) (cid:48)(cid:48) : ρ ( S ) = 1 , ρ ( T ) = ω , (B.9) : ρ ( S ) = 13 − − − , ρ ( T ) = ω
00 0 ω , (B.10)where ω = e πi/ . B.2.3 Γ (cid:39) S : ρ ( S ) = 1 , ρ ( T ) = 1 , (B.11) (cid:48) : ρ ( S ) = − , ρ ( T ) = − , (B.12) : ρ ( S ) = 12 (cid:18) − √ √ (cid:19) , ρ ( T ) = (cid:18) − (cid:19) , (B.13) : ρ ( S ) = − √ √ √ − √ − , ρ ( T ) = − i
00 0 − i , (B.14) (cid:48) : ρ ( S ) = 12 √ √ √ − √ − , ρ ( T ) = i
00 0 − i . (B.15) B.2.4 Γ (cid:39) A : ρ ( S ) = 1 , ρ ( T ) = 1 , (B.16) : ρ ( S ) = 1 √ −√ −√ −√ − ϕ /ϕ −√ /ϕ − ϕ , ρ ( T ) = ζ
00 0 ζ , (B.17) (cid:48) : ρ ( S ) = 1 √ − √ √ √ − /ϕ ϕ √ ϕ − /ϕ , ρ ( T ) = ζ
00 0 ζ , (B.18) : ρ ( S ) = 1 √ /ϕ ϕ − /ϕ − ϕϕ − /ϕ − ϕ /ϕ , ρ ( T ) = ζ ζ ζ
00 0 0 ζ , (B.19) : ρ ( S ) = 15 − √ √ √ √ √ /ϕ − ϕ /ϕ ϕ √ − ϕ ϕ /ϕ /ϕ √ /ϕ /ϕ ϕ − ϕ √ ϕ /ϕ − ϕ /ϕ , ρ ( T ) = ζ ζ ζ
00 0 0 0 ζ , (B.20)where ζ = e πi/ and ϕ = (1 + √ /
2. 23 .3 Clebsch-Gordan Coefficients
For completeness, and for each level N = 2 , , ,
5, we also reproduce here the nontrivialClebsch-Gordan coefficients in the symmetric basis of Appendix B.2. Entries of each multipletentering the tensor product are denoted by α i and β i . B.3.1 Γ (cid:39) S (cid:48) ⊗ (cid:48) = ∼ α β (cid:48) ⊗ = ∼ (cid:18) − α β α β (cid:19) (B.21) ⊗ = ⊕ (cid:48) ⊕ ∼ α β + α β (cid:48) ∼ α β − α β ∼ (cid:18) α β + α β α β − α β (cid:19) (B.22) B.3.2 Γ (cid:39) A (cid:48) ⊗ (cid:48) = (cid:48)(cid:48) ∼ α β (cid:48)(cid:48) ⊗ (cid:48)(cid:48) = (cid:48) ∼ α β (cid:48) ⊗ (cid:48)(cid:48) = ∼ α β (cid:48) ⊗ = ∼ α β α β α β (cid:48)(cid:48) ⊗ = ∼ α β α β α β (B.23) ⊗ = ⊕ (cid:48) ⊕ (cid:48)(cid:48) ⊕ ⊕ ∼ α β + α β + α β (cid:48) ∼ α β + α β + α β (cid:48)(cid:48) ∼ α β + α β + α β ∼ α β − α β − α β α β − α β − α β α β − α β − α β ∼ α β − α β α β − α β α β − α β (B.24)24 .3.3 Γ (cid:39) S (cid:48) ⊗ (cid:48) = ∼ α β (cid:48) ⊗ = ∼ (cid:18) α β − α β (cid:19) (cid:48) ⊗ = (cid:48) ∼ α β α β α β (cid:48) ⊗ (cid:48) = ∼ α β α β α β (B.25) ⊗ = ⊕ (cid:48) ⊕ ∼ α β + α β (cid:48) ∼ α β − α β ∼ (cid:18) α β − α β α β + α β (cid:19) ⊗ = ⊕ (cid:48) ∼ α β (cid:0) √ / (cid:1) α β − (1 / α β (cid:0) √ / (cid:1) α β − (1 / α β (cid:48) ∼ − α β (cid:0) √ / (cid:1) α β + (1 / α β (cid:0) √ / (cid:1) α β + (1 / α β ⊗ (cid:48) = ⊕ (cid:48) ∼ − α β (cid:0) √ / (cid:1) α β + (1 / α β (cid:0) √ / (cid:1) α β + (1 / α β (cid:48) ∼ α β (cid:0) √ / (cid:1) α β − (1 / α β (cid:0) √ / (cid:1) α β − (1 / α β (B.26)25 ⊗ = (cid:48) ⊗ (cid:48) = ⊕ ⊕ ⊕ (cid:48) ∼ α β + α β + α β ∼ (cid:18) α β − (1 /
2) ( α β + α β ) (cid:0) √ / (cid:1) ( α β + α β ) (cid:19) ∼ α β − α β α β + α β − α β − α β (cid:48) ∼ α β − α β α β − α β α β − α β (B.27) ⊗ (cid:48) = (cid:48) ⊕ ⊕ ⊕ (cid:48) (cid:48) ∼ α β + α β + α β ∼ (cid:18) (cid:0) √ / (cid:1) ( α β + α β ) − α β + (1 /
2) ( α β + α β ) (cid:19) ∼ α β − α β α β − α β α β − α β (cid:48) ∼ α β − α β α β + α β − α β − α β (B.28) B.3.4 Γ (cid:39) A ⊗ = ⊕ ⊕ ∼ α β + α β + α β ∼ α β − α β α β − α β α β − α β ∼ α β − α β − α β −√ α β − √ α β √ α β √ α β −√ α β − √ α β (B.29)26 ⊗ (cid:48) = ⊕ ∼ √ α β + α β −√ α β − α β −√ α β − α β √ α β + α β ∼ √ α β α β − √ α β α β − √ α β α β − √ α β α β − √ α β ⊗ = (cid:48) ⊕ ⊕ (cid:48) ∼ −√ α β − √ α β √ α β − α β + α β √ α β + α β − α β ∼ α β − √ α β − α β − √ α β α β + √ α β − α β + √ α β ∼ √ α β − √ α β √ α β + 2 α β −√ α β + α β + 3 α β √ α β − α β − α β − √ α β − α β ⊗ = ⊕ (cid:48) ⊕ ⊕ ∼ − α β + √ α β + √ α β √ α β + α β − √ α β √ α β − √ α β + α β (cid:48) ∼ √ α β + α β + α β α β − √ α β − √ α β α β − √ α β − √ α β ∼ √ α β − √ α β + α β −√ α β + 2 α β − α β √ α β + 3 α β − α β − √ α β − α β + √ α β ∼ √ α β − √ α β − α β − √ α β − √ α β − α β − √ α β α β + √ α β α β + √ α β + √ α β (B.30)27 (cid:48) ⊗ (cid:48) = ⊕ (cid:48) ⊕ ∼ α β + α β + α β (cid:48) ∼ α β − α β α β − α β α β − α β ∼ α β − α β − α β √ α β −√ α β − √ α β −√ α β − √ α β √ α β (cid:48) ⊗ = ⊕ ⊕ ∼ −√ α β − √ α β √ α β + α β − α β √ α β − α β + α β ∼ α β + √ α β α β − √ α β − α β + √ α β − α β − √ α β ∼ √ α β − √ α β √ α β − α β − α β √ α β + 2 α β − √ α β − α β −√ α β + α β + 3 α β (cid:48) ⊗ = ⊕ (cid:48) ⊕ ⊕ ∼ √ α β + α β + α β α β − √ α β − √ α β α β − √ α β − √ α β (cid:48) ∼ − α β + √ α β + √ α β √ α β + α β − √ α β √ α β − √ α β + α β ∼ √ α β + 3 α β − α β √ α β − √ α β + α β − √ α β − α β + √ α β −√ α β + 2 α β − α β ∼ √ α β − √ α β α β + √ α β − α β − √ α β − √ α β α β + √ α β + √ α β − α β − √ α β (B.31)28 ⊗ = ⊕ ⊕ (cid:48) ⊕ ⊕ ∼ α β + α β + α β + α β ∼ − α β + α β − α β + α β √ α β − √ α β √ α β − √ α β (cid:48) ∼ α β + α β − α β − α β √ α β − √ α β √ α β − √ α β ∼ α β + α β + α β α β + α β + α β α β + α β + α β α β + α β + α β ∼ √ α β − √ α β − √ α β + √ α β −√ α β + 2 √ α β − √ α β − √ α β + √ α β + √ α β √ α β + √ α β − √ α β −√ α β + 2 √ α β − √ α β ⊗ = ⊕ (cid:48) ⊕ ⊕ ⊕ ∼ √ α β − √ α β + √ α β − √ α β −√ α β + 2 α β + 3 α β − α β α β − α β − α β + √ α β (cid:48) ∼ √ α β + 2 √ α β − √ α β − √ α β α β − √ α β − α β + 2 α β − α β + α β + √ α β − α β ∼ √ α β − √ α β + √ α β − √ α β −√ α β − √ α β + 2 √ α β + √ α β √ α β + 2 √ α β − √ α β − √ α β − √ α β + √ α β − √ α β + √ α β ∼ √ α β − √ α β − √ α β + √ α β −√ α β − √ α β − √ α β √ α β + √ α β + √ α β √ α β + √ α β + √ α β −√ α β − √ α β − √ α β ∼ α β + 4 α β + 4 α β + 2 α β α β + 2 √ α β −√ α β + 2 α β − √ α β + 2 √ α β √ α β − √ α β + 2 α β − √ α β √ α β + 4 α β (B.32)29 ⊗ = ⊕ ⊕ (cid:48) ⊕ ⊕ ⊕ ⊕ ∼ α β + α β + α β + α β + α β ∼ α β + 2 α β − α β − α β −√ α β + √ α β + √ α β − √ α β √ α β + √ α β − √ α β − √ α β (cid:48) ∼ α β − α β + α β − α β √ α β − √ α β + √ α β − √ α β −√ α β + √ α β − √ α β + √ α β ∼ √ α β + 3 √ α β − √ α β + 4 √ α β − √ α β √ α β + 4 √ α β + 3 √ α β − √ α β − √ α β √ α β − √ α β − √ α β + 3 √ α β + 4 √ α β √ α β − √ α β + 4 √ α β − √ α β + 3 √ α β ∼ √ α β − √ α β + √ α β − √ α β −√ α β + √ α β + √ α β − √ α β −√ α β − √ α β + √ α β + √ α β √ α β − √ α β + √ α β − √ α β ∼ α β + α β − α β − α β + α β α β + α β + √ α β + √ α β − α β + √ α β − α β − α β − α β + √ α β α β + √ α β + √ α β + α β ∼ α β − α β + α β + α β − α β − α β − α β + √ α β α β + α β + √ α β + √ α β α β + √ α β + √ α β + α β − α β + √ α β − α β (B.33) C Modular Form Multiplets
C.1 Lowest Weight Multiplets
For the groups Γ N with N ≤
5, the lowest (non-trivial) weight modular multiplets can beconstructed from linear combinations Y ( a i | τ ) of logarithmic derivatives of some seed func-tions. These functions are the Dedekind eta η ( τ ) and Jacobi theta θ ( z ( τ ) , t ( τ )) functions, with modified arguments. The Dedekind eta is defined as an infinite product, η ( τ ) = q / ∞ (cid:89) n =1 (1 − q n ) , q = e πiτ , (C.1)while the Jacobi θ can be written as the series θ ( z, τ ) = ∞ (cid:88) n = −∞ ˜ q n e πinz , ˜ q = e πiτ . (C.2) For the properties of this last special function, see, e.g. [49, 50]. In the notations of Ref. [49], θ ≡ θ (cid:20) (cid:21) . N Γ (cid:39) S Γ (cid:39) A Γ (cid:39) S Γ (cid:39) A Linear space dim. at weight k k/ k + 1 2 k + 1 5 k + 1Linear space dim. at weight 2 2 3 5 11Lowest weight irreps , (cid:48) , (cid:48) , Table 3:
Dimensions of the linear spaces of modular forms at generic weight k and at thelowest weight ( k = 2) [51], including the irrep breakdown at the lowest weight [21, 23, 30, 32],for the finite modular groups Γ N with N ≤ Below, for each N ≤
5, we reproduce explicit expressions for the lowest weight modularmultiplets Y ( N,k =2) r as vectors of the aforementioned log-derivatives. Note that these multi-plets are given in the symmetric basis of Appendix B.2. For these groups, the dimensions oflinear spaces at different weights and the irreducible representations arising at lowest weightare summarised in Table 3. C.1.1 Γ (cid:39) S The lowest weight multiplet for Γ was derived in Ref. [23], and reads (up to normalisation): Y (2 , ( τ ) = i (cid:18) Y (2) (1 , , − | τ ) Y (2) ( √ , −√ , | τ ) (cid:19) , (C.3)with Y (2) ( a , a , a | τ ) = dd τ (cid:20) a log η (cid:16) τ (cid:17) + a log η (cid:18) τ + 12 (cid:19) + a log η (2 τ ) (cid:21) . (C.4) C.1.2 Γ (cid:39) A The lowest weight multiplet for Γ was derived in Ref. [21], and reads (up to normalisation): Y (3 , ( τ ) = i Y (3) (1 / , / , / , − / | τ ) − Y (3) (1 , ω , ω, | τ ) − Y (3) (1 , ω, ω , | τ ) , (C.5)with ω = e πi/ and Y (3) ( a , . . . , a | τ ) = dd τ (cid:20) a log η (cid:16) τ (cid:17) + a log η (cid:18) τ + 13 (cid:19) + a log η (cid:18) τ + 23 (cid:19) + a log η (3 τ ) (cid:21) . (C.6) C.1.3 Γ (cid:39) S The lowest weight multiplets for Γ were derived in Ref. [30], in a non-symmetric basis forthe representation of group generators. In the symmetric basis we here consider, they read(up to normalisation): Y (4 , ( τ ) = i (cid:18) Y (4) (1 , , − / , − / , − / , − / | τ ) Y (4) (0 , , √ / , −√ / , √ / , −√ / | τ ) (cid:19) , (C.7)31 (4 , (cid:48) ( τ ) = i Y (4) (1 , − , , , , | τ ) Y (4) (0 , , − / √ , i/ √ , / √ , − i/ √ | τ ) Y (4) (0 , , − / √ , − i/ √ , / √ , i/ √ | τ ) , (C.8)with Y (4) ( a , . . . , a | τ ) = dd τ (cid:20) a log η (cid:18) τ + 12 (cid:19) + a log η (4 τ ) + a log η (cid:16) τ (cid:17) + a log η (cid:18) τ + 14 (cid:19) + a log η (cid:18) τ + 24 (cid:19) + a log η (cid:18) τ + 34 (cid:19) (cid:21) . (C.9) C.1.4 Γ (cid:39) A The lowest weight multiplets for Γ were derived in Ref. [32], and read (up to normalisation): Y (5 , ( τ ) = i − √ Y (5) ( − , , , , , − , , , , , | τ ) Y (5) (0 , , ζ , ζ , ζ , ζ ; 0 , , ζ , ζ , ζ , ζ | τ ) Y (5) (0 , , ζ , ζ, ζ , ζ ; 0 , , ζ , ζ, ζ , ζ | τ ) Y (5) (0 , , ζ , ζ , ζ, ζ ; 0 , , ζ , ζ , ζ, ζ | τ ) Y (5) (0 , , ζ, ζ , ζ , ζ ; 0 , , ζ, ζ , ζ , ζ | τ ) , (C.10) Y (5 , ( τ ) = i √ Y (5) (cid:0) −√ , − , − , − , − , − √ , , , , , (cid:12)(cid:12) τ (cid:1) Y (5) (0 , , ζ , ζ , ζ , ζ ; 0 , − , − ζ , − ζ , − ζ , − ζ | τ ) Y (5) (0 , , ζ, ζ , ζ , ζ ; 0 , − , − ζ, − ζ , − ζ , − ζ | τ ) , (C.11) Y (5 , (cid:48) ( τ ) = i √ Y (5) (cid:0) √ , − , − , − , − , − −√ , , , , , (cid:12)(cid:12) τ (cid:1) Y (5) (0 , , ζ , ζ, ζ , ζ ; 0 , − , − ζ , − ζ, − ζ , − ζ | τ ) Y (5) (0 , , ζ , ζ , ζ, ζ ; 0 , − , − ζ , − ζ , − ζ, − ζ | τ ) , (C.12)with ζ = e πi/ and Y (5) ( c , − , . . . , c , ; c , − , . . . , c , | τ ) = (cid:88) i,j c i,j dd τ log α i,j ( τ ) , (C.13)the corresponding seed functions being α , − ( τ ) = θ (cid:18) τ + 12 , τ (cid:19) ,α , ( τ ) = θ (cid:18) τ + 910 , τ (cid:19) ,α , ( τ ) = θ (cid:18) τ , τ + 15 (cid:19) ,α , ( τ ) = θ (cid:18) τ + 110 , τ + 25 (cid:19) ,α , ( τ ) = θ (cid:18) τ + 210 , τ + 35 (cid:19) ,α , ( τ ) = θ (cid:18) τ + 310 , τ + 45 (cid:19) , α , − ( τ ) = e πiτ/ θ (cid:18) τ + 12 , τ (cid:19) ,α , ( τ ) = θ (cid:18) τ + 710 , τ (cid:19) ,α , ( τ ) = θ (cid:18) τ + 810 , τ + 15 (cid:19) ,α , ( τ ) = θ (cid:18) τ + 910 , τ + 25 (cid:19) ,α , ( τ ) = θ (cid:18) τ , τ + 35 (cid:19) ,α , ( τ ) = θ (cid:18) τ + 110 , τ + 45 (cid:19) . (C.14)32 .2 Bases for Spaces of Lowest Weight Forms and Their q -Expansions For each Γ N , one can obtain q -expansions for a basis b ( N ) i (the so-called Miller-like basis)of the space of lowest weight modular forms from the SageMath algebra system [52] (see,e.g., [32]). The dimensions of these linear spaces have been given in Table 3. Below, foreach N ≤
5, we present these expansions as well as the decompositions of the lowest weightmultiplets of Appendix C.1 (given in the symmetric basis of Appendix B.2) in terms of thebasis vectors b ( N ) i ≡ b ( N ) i ( τ ). C.2.1 Γ (cid:39) S For Γ , the Miller-like basis reads: b (2)1 = 1 + 24 q + 24 q + 96 q + 24 q + 144 q + 96 q + 192 q + 24 q + 312 q + 144 q + . . . ,b (2)2 = q + 4 q + 6 q + 8 q + 13 q + 12 q + 14 q + 24 q + 18 q + 20 q + . . . , (C.15)with q ≡ e πi τ/ = e πi τ . The lowest weight modular multiplet of Γ can be written in termsof the above vectors, namely (see also [23]): Y (2 , ( τ ) = 2 π (cid:32) b (2)1 ( τ ) / √ b (2)2 ( τ ) (cid:33) . (C.16)In the present Appendix, modular multiplets are properly normalised, guaranteeing Y ( − τ ∗ ) = Y ∗ ( τ ). C.2.2 Γ (cid:39) A For Γ , the Miller-like basis reads: b (3)1 = 1 + 12 q + 36 q + 12 q + 84 q + 72 q + 36 q + 96 q + 180 q + 12 q + 216 q + . . . ,b (3)2 = q + 7 q + 8 q + 18 q + 14 q + 31 q + 20 q + 36 q + 31 q + 56 q + . . . ,b (3)3 = q + 2 q + 5 q + 4 q + 8 q + 6 q + 14 q + 8 q + 14 q + 10 q + . . . , (C.17)with q ≡ e πi τ/ . The lowest weight modular multiplet of Γ can be written in terms of theabove vectors, namely (see also [21]): Y (3 , ( τ ) = π b (3)1 ( τ ) − b (3)2 ( τ ) − b (3)3 ( τ ) . (C.18)33 .2.3 Γ (cid:39) S For Γ , the Miller-like basis reads: b (4)1 = 1 + 24 q + 24 q + 96 q + 24 q + 144 q + . . . ,b (4)2 = q + 6 q + 13 q + 14 q + 18 q + 32 q + 31 q + 30 q + 48 q + 38 q + . . . ,b (4)3 = q + 4 q + 6 q + 8 q + 13 q + 12 q + 14 q + 24 q + 18 q + 20 q + . . . ,b (4)4 = q + 2 q + 3 q + 6 q + 5 q + 6 q + 10 q + 8 q + 12 q + 14 q + . . . ,b (4)5 = q + 4 q + 6 q + 8 q + 13 q + . . . , (C.19)with q ≡ e πi τ/ = e πi τ/ . The lowest weight modular multiplets of Γ can be written interms of the above vectors, namely: Y (4 , ( τ ) = − π (cid:32) b (4)1 ( τ ) / b (4)5 ( τ ) −√ b (4)3 ( τ ) (cid:33) , (C.20) Y (4 , (cid:48) ( τ ) = − π − b (4)1 ( τ ) / b (4)5 ( τ ) √ b (4)2 ( τ )4 √ b (4)4 ( τ ) . (C.21) C.2.4 Γ (cid:39) A For Γ , the Miller-like basis reads: b (5)1 = 1 + 60 q − q + 240 q − q + 300 q − q + 240 q + . . . ,b (5)2 = q + 12 q + 7 q + 8 q + 6 q + 32 q + 7 q + 42 q + 12 q + . . . ,b (5)3 = q + 12 q − q + 12 q + 8 q + 21 q − q + 48 q − q + . . . ,b (5)4 = q + 11 q − q + 21 q − q + 12 q + 41 q − q + . . . ,b (5)5 = q + 9 q − q + 29 q − q + 17 q + 8 q + 12 q − q + . . . ,b (5)6 = q + 6 q − q + 27 q − q + 30 q − q + 26 q + . . . ,b (5)7 = q + 2 q + 2 q + 3 q + 7 q + 5 q + . . . ,b (5)8 = q − q + 3 q + 2 q + 7 q − q + 9 q + . . . ,b (5)9 = q − q + 5 q − q + 4 q + 4 q − q + 16 q + . . . ,b (5)10 = q − q + 8 q − q + 12 q − q + 13 q + . . . ,b (5)11 = q − q + 12 q − q + 30 q − q + 5 q + 18 q − q + . . . , (C.22)with q ≡ e πi τ/ . The lowest weight modular multiplets of Γ can be written in terms of theabove vectors, namely (see also [32]): Y (5 , ( τ ) = − √ π (cid:16) b (5)1 ( τ ) − b (5)6 ( τ ) − b (5)11 ( τ ) (cid:17) / (5 √ b (5)2 ( τ ) + 2 b (5)7 ( τ )3 b (5)5 ( τ ) + 7 b (5)10 ( τ ) , (C.23)34 (5 , (cid:48) ( τ ) = − √ π − (cid:16) b (5)1 ( τ ) + 20 b (5)6 ( τ ) + 30 b (5)11 ( τ ) (cid:17) / (5 √ b (5)3 ( τ ) + 6 b (5)8 ( τ )2 b (5)4 ( τ ) + 5 b (5)9 ( τ ) , (C.24) Y (5 , ( τ ) = 2 π − (cid:16) b (5)1 ( τ ) + 6 b (5)6 ( τ ) + 18 b (5)11 ( τ ) (cid:17) / √ b (5)2 ( τ ) + 12 b (5)7 ( τ )3 b (5)3 ( τ ) + 8 b (5)8 ( τ )4 b (5)4 ( τ ) + 15 b (5)9 ( τ )7 b (5)5 ( τ ) + 13 b (5)10 ( τ ) . (C.25) C.3 Higher Weight Multiplets for Γ (cid:39) S in the Symmetric Basis Multiplets of higher weight Y ( N,k> r ,µ , with the index µ labelling linearly independent multi-plets, may be obtained from those of lower weight via tensor products. For Γ , multiplets ofweight up to 6 are found in Ref. [21], while for Γ and Γ , multiplets of weight up to 10 aregiven in Refs. [31] and [32], respectively.Since in [31] the considered basis for Γ is not a symmetric one, we here present explicitexpressions for higher weight multiplets in the symmetric basis of Appendix B.2. Making useof the shorthands Y (4 , = ( Y , Y ) T and Y (4 , (cid:48) = ( Y , Y , Y ) T , one has at weight 4, Y (4 , = Y + Y , Y (4 , = (cid:18) Y − Y Y Y (cid:19) ,Y (4 , = − Y Y √ Y Y + Y Y √ Y Y + Y Y , Y (4 , (cid:48) = Y Y √ Y Y − Y Y √ Y Y − Y Y , (C.26)at weight 6, Y (4 , = Y (cid:0) Y − Y (cid:1) , Y (4 , (cid:48) = Y (cid:0) Y − Y (cid:1) ,Y (4 , = (cid:0) Y + Y (cid:1) (cid:18) Y Y (cid:19) , Y (4 , = Y (cid:0) Y − Y (cid:1) Y (cid:0) Y Y + √ Y Y (cid:1) − Y (cid:0) Y Y + √ Y Y (cid:1) ,Y (4 , (cid:48) , = (cid:0) Y + Y (cid:1) Y Y Y , Y (4 , (cid:48) , = Y (cid:0) Y − Y (cid:1) − Y (cid:0) Y Y − √ Y Y (cid:1) Y (cid:0) Y Y − √ Y Y (cid:1) , (C.27)35t weight 8, Y (4 , = (cid:0) Y + Y (cid:1) ,Y (4 , , = (cid:0) Y + Y (cid:1) (cid:18) Y − Y Y Y (cid:19) , Y (4 , , = (cid:0) Y − Y (cid:1) (cid:18) Y Y Y (cid:19) ,Y (4 , , = (cid:0) Y + Y (cid:1) − Y Y Y Y + √ Y Y Y Y + √ Y Y , Y (4 , , = Y (cid:0) Y − Y (cid:1) Y Y Y ,Y (4 , (cid:48) , = (cid:0) Y + Y (cid:1) − Y Y Y Y − √ Y Y Y Y − √ Y Y , Y (4 , (cid:48) , = Y (cid:0) Y − Y (cid:1) Y Y Y , (C.28)and at weight 10, Y (4 , = Y (cid:0) Y − Y (cid:1) (cid:0) Y + Y (cid:1) , Y (4 , (cid:48) = Y (cid:0) Y − Y (cid:1) (cid:0) Y + Y (cid:1) ,Y (4 , , = (cid:0) Y + Y (cid:1) (cid:18) Y Y (cid:19) , Y (4 , , = (cid:0) Y − Y (cid:1) (cid:18) Y − Y Y − Y Y (cid:19) ,Y (4 , , = Y (cid:0) Y − Y (cid:1) − Y Y Y Y + √ Y Y Y Y + √ Y Y , Y (4 , , = Y (cid:0) Y − Y (cid:1) − Y Y Y Y − √ Y Y Y Y − √ Y Y ,Y (4 , (cid:48) , = Y (cid:0) Y − Y (cid:1) − Y Y Y Y − √ Y Y Y Y − √ Y Y , Y (4 , (cid:48) , = Y (cid:0) Y − Y (cid:1) − Y Y Y Y + √ Y Y Y Y + √ Y Y ,Y (4 , (cid:48) , = (cid:0) Y + Y (cid:1) Y Y Y . 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