Modular invariant holomorphic observables

In modular invariant models of flavor, observables must be modular invariant. The observables discussed so far in the literature are functions of the modulus $\tau$ and its conjugate, $\bar\tau$. We point out that certain combinations of observables depend only on $\tau$, i.e. are meromorphic, and in some cases even holomorphic functions of $\tau$. These functions, which we dub ``invariants'' in this Letter, are highly constrained, renormalization group invariant, and allow us to derive many of the models' features without the need for extensive parameter scans. We illustrate the robustness of these invariants in two existing models in the literature based on modular symmetries, $\Gamma_{3}$ and $\Gamma_{5}$. We find that, in some cases, the invariants give rise to robust relations among physical observables that are independent of $\tau$. Furthermore, there are instances where additional symmetries exist among the invariants. These symmetries are relevant phenomenologically and may provide a dynamical way to realize symmetries of mass matrices.

One of the main reasons for the popularity of this scheme is that the couplings of the theory are unique, or at least very constrained.In slightly more detail, the couplings are functions of a chiral superfield  subject to three requirements: modular covariance or modular invariance (cf.Section 2), τ the couplings depend only on the modulus  but not on its conjugate, τ, and ∞ the couplings are finite for all values of .
Note that in different communities, different terminology is being used for these requirements.In mathematics, τ amounts to saying that the coupling is a meromorphic function of  whereas in some physics contexts such functions are called holomorphic.
τ and ∞ together mean, in mathematician's terminology, that the coupling is a holomorphic function of .In what follows, we will refer to the requirements just by the symbols to avoid confusion.
The important point is that , τ and ∞ together are so restrictive that they almost completely fix the couplings [18][19][20].Up to this point, these requirements have only been discussed at the level of superpotential couplings.The purpose of this Letter is to point out that one can make similar statements at the level of observables.As we shall see, there are observables for which all the requirements, i.e. , τ and ∞ , are simultaneously fulfilled.Our findings allow us to make very robust predictions.There are also observables which fulfill and τ but fail to satisfy ∞ .
We will comment on how in such cases one can still make general statements on the predictions of the model.
In this Letter, we first review the basic framework of modular flavor symmetries in Section 2. In Section 3 we discuss how typical observables are non-holomorphic since they involve the normalization of the fields.We then introduce modular invariant holomorphic observables in Section 4, and work out some basic applications using two example models in Sections 4.1 and 4.2.Section 5 contains some further discussion, and Section 6 contains our conclusions.

A short recap of modular flavor symmetries
The key ingredient of modular flavor symmetries is modular invariance, i.e. requirement .That is, the theory is assumed to be invariant under SL(2, Z) transformations  of the modulus , where , , ,  ∈ Z and   −   = 1.Modular invariance, along with τ from supersymmetry (SUSY)1 and the additional requirement that the couplings of the theory be finite, i.e. ∞ , leads to a 1SUSY may not be necessary for τ , cf. [21].However, so far there is no explicit model illustrating this.
highly predictive scheme, in which the superpotential couplings are almost unique.That is, the superpotential terms of the models are of the form where    () are uniquely determined vector-valued modular forms and the Φ  denote some appropriate superfields.That is, under (1)    ()  ↦ − − →    ( ) = ( + )     ()   () , where   is a representation matrix of a finite group.As long as   ≠ 0 and/or   () ≠ 1,   () is modular covariant (rather than invariant).The superfields transform as where   denotes the modular weight of Φ  , which, as indicated may transform nontrivially under the finite modular group with representation matrix   (). denotes a coefficient, which can be chosen arbitrarily in the bottom-up approach.2However, apart from this freedom, the superpotential terms are uniquely determined by requirements , τ and ∞ .
Let us briefly recall what ∞ means.It is the requirement that the functions   () remains finite throughout the fundamental domain.Without this requirement, we could multiply   () by arbitrary polynomials of the modular invariant function, or Hauptmodul of SL(2, Z), (), while still satisfying requirements and τ .However, as  diverges for  → i ∞, this is inconsistent with ∞ , and therefore not allowed.Thus, by requiring simultaneously , τ , and ∞ , the couplings are unique up to an undetermined coefficient , and in cases where there are multiple invariant contractions, up to multiple undetermined coefficients   .Examples for the latter case can be found in [4,7,13].
In vast literature, the Kähler potential of the matter fields is assumed to be of the so-called minimal form, Here we set the vector multiplets to zero.It is known that the requirements , τ and ∞ do not fix the Kähler potential to be of the form (5), but there are several additional terms which are allowed by the symmetries of the models, thus limiting the predictive power of the models [24].While entirely convincing solutions to this problem have not yet been found, there exist proof-of-principle type fixes which allow one to sufficiently control the extra terms to make their impact comparable to the current experimental uncertainties in flavor observables [25].In what follows, we will base our discussion on the minimal Kähler potential (5).Modular invariance, in particular, means that observables are to be modular invariant.However, this does not mean that the    () of ( 2) are modular invariant.Rather, as we shall see next, there are several non-holomorphic observables which are modular invariant because of the normalization of the fields, cf.Equation (5).
2See, however, [22] for a proposal for the normalization of the modular forms.It would be interesting to see to which extent this approach replicates the known normalizations in explicit top-down constructions such as [23].

Non-holomorphic observables
To illustrate this point with an explicit example, we consider a toy model based on where ℳ() is a vector-valued modular form of weight  ℳ .Apart from the modular weight of the field,  Φ , we need to specify the modular weight of the superpotential  W =  ℳ + 2 Φ .In large parts of the literature, the modular weight of the superpotential is taken to be zero,  W = 0, and in this section we adopt this practice.A nonzero  W = 0, as required by supergravity, does not change the following discussion qualitatively.This then fixes the modular weight of ℳ() to be Thus, under a modular transformation While ℳ() and  both transform nontrivially, is straightforward to confirm that is invariant under (8).This combination emerges from the scalar potential, after rescaling the fields to be canonically normalized.That is, we have to take into account the inverse of the Kähler metric, which we obtain from Equation (6b) both in (10) and when computing the physical mass, Here, Φ denotes the scalar component of the superfield Φ.The resulting physical mass of Φ is given by As indicated by the notation, the physical mass is not a meromorphic (nor holomorphic) function of , i.e. it does not fulfill τ .Of course, the physical mass  physical ( τ, ) is modular invariant, as it should, i.e. satisfies .However, it is modular invariant "at the expense" of being non-holomorphic, i.e. and ∞ are fulfilled but not τ .
The observable of the model, i.e. the mass, fails to satisfy τ because it involves the Kähler metric.
Let us stress that this feature of the toy model is rather generic: in order to compute observables, one typically needs to take into account the Kähler metrics, thereby sacrificing τ .As a consequence, the uniqueness discussed in Section 2 does not apply to such observables.In what follows, we will see that there are observables that do not receive τ-dependents contributions, and hence fulfill τ , and are thus highly constrained.

Modular invariant holomorphic observables
In order to obtain holomorphic observables, we need to remove the nonholomorphic terms coming from the Kähler metric.It turns out that in the lepton sector of the minimal supersymmetric standard model (MSSM) there is a straightforward way to obtain such expressions.Consider the superpotential of the lepton sector Here,   and   denote the three generations of the superfields of the SU(2) L charged lepton doublets and singlets, and  / stand for the MSSM Higgs doublets.  denotes the charged lepton Yukawa couplings, which is not a modular form.() =  2  () is the neutrino mass matrix, with () being the effective neutrino mass operator.
In the basis in which   = diag(  ,   ,   ), consider where no summation over ,  is implied.Here,   (, τ) • • = (− i  + i τ) (   +   )/2   ()  2  are the entries of the neutrino mass matrix in the canonically normalized basis, with    being the modular weight of the lepton doublet   .Crucially, Equation (15) shows that the ratios of the physical mass matrix entries,   (, τ), can be expressed entirely as rational functions of holomorphic modular functions.This is because, while the individual entries   (, τ) have a structure analogous to Equation (13), the   are constructed in such a way that the factors containing τ cancel.That is, by construction the   fulfill τ .In what follows, we will discuss to which extent they also fulfill and ∞ .It has been known for a while that the   from Equation ( 15) are renormalization group (RG) invariant [26] (see the discussion in Appendix A.) In the MSSM this can be understood from the non-renormalization theorem and the fact that the normalizations of the field cancel, which is also the reason why the   from Equation ( 15) are interesting for our present discussion.As we detail in Appendix A, the location of zeros and poles are RG invariant to all orders even in the absence of SUSY.In this basis, the neutrino mass matrix is given by where the   denote the neutrino mass eigenvalues and the PMNS matrix  PMNS depends on the leptonic mixing angles ( 12 ,  13 ,  23 ), the Dirac phase () and the two Majorana phases ( 1 ,  2 ).Altogether, there are nine independent physical parameters, From Equation ( 16), the invariants can be computed explicitly, and read, in the PDG basis,3 where The invariants   depend on  1 ,  2 ,  1 and  2 only via the combinations  1 • • =  1 e i  1 and As   are complex constants, they give rise to six relations.Based on these relations, one can infer, for instance, the scale dependence of all angles and phases from the running of the three mass eigenvalues.While the expressions (18) are lengthy, they have two important properties: 1. they only depend on the physical parameters (17); 2. they are modular invariant.
In models in which the Majorana neutrino masses are modular forms, the modular weights of the matrix elements of the light neutrino mass matrix are solely determined by the modular weights of the left-handed leptons   .Then the matrix element   () of the superpotential coupling matrix () has modular weight    +    .The invariants (15) must be modular functions of weight 2   + 2   − 2(   +    ) = 0 of the corresponding modular symmetry.This means that the   () can always be written as rational functions of corresponding so-called Hauptmodul of the corresponding modular symmetry [19,20].The Hauptmodul for a given subgroup  of SL(2, Z) is a modular function of weight 0 on  which generates all the modular functions for this group , and fulfills and τ .The best-known example of this kind is the -invariant () • • =  3 4 ()/ 24 (), which is the Hauptmodul for the full modular group SL(2, Z).Here  4 () is the Eisenstein series and () is the Dedekind eta function.Notice that these modular invariant functions have, as opposed to the modular forms, poles [28], i.e. they fulfill & τ but not ∞ .Given these properties we have a significant amount of information on these physical observables directly from the theory of modular forms.We will illustrate this crucial point in the following examples.

Feruglio Model based on 𝚪 3
Consider Model 1 from [1], which is based on finite modular group Γ 3  4 .The assignments of modular weights and representations for the matter fields are shown in (2) contains a triplet flavon,   , which only couples to the charged leptons.The effective neutrino masses depend only on the modular forms of weight 2. In more detail, the relevant terms of the superpotential are given by and there are no higher-order contributions to the effective neutrino mass operator in the superpotential.The notation (. . . )  indicates a contraction to the representation  of the finite group, i.e.  4 in this case, and does not imply the SL(2, Z) representation explicitly.The flavon   is assumed to develop the vacuum expectation value (VEV) With the VEV given in (21), the charged lepton and neutrino mass matrices read Here,  1,2,3 are the components of modular forms triplet (2) 3 ().Notice that  <  < , and as a consequence  2 and  3 in (22b) are swapped compared to [1,Equation (38)].
The invariants (15) are given by The invariants   () are products of ratios of two holomorphic modular forms, so they are meromorphic on the extended upper-half plane ℋ • • = ℋ ∪ R ∪ {i ∞}.They also transform as  4 1-plets.Consider a modular invariant meromorphic function ℐ().Modular invariance, as opposed to modular covariance, means that (cf.e.g.[28]) or it has poles.
Both cases are realized in the example at hand.First of all,  12 is a constant because the   satisfy the algebraic constraint4 The latter follows from the fact that the modular form triplet 3 () of weight 2 can be obtained from the tensor product of modular form doublet  (1) Here, the modular forms  1,2 () of weight 1 on Γ(3) are given by where Since the three   can be expressed in terms of the two   , the   are not algebraically independent, as manifest in the constraint in Equation (24).
The conditions in Equations (28a) to (28c) lead to robust phenomenological implications when relations in Equation ( 18) are utilized where the invariants are expressed in terms of the physical mixing parameters (17).In particular, Equation (28a) and Equation (29) give rise to four constraints that are independent of the value of .Due to the form of the neutrino mass matrix in this model Equation (22b), there is a sum rule among the three physical neutrino masses [29,30],5 Given the sum rule, the three neutrino masses (and thus the absolute neutrino mass scale) are completely fixed by the two mass squared differences, Δ 2 sol and Δ 2 atm , which have been determined from oscillation experiments.Furthermore, the mixing angles are also known from oscillation experiments [32].We are thus left with three undetermined observables, namely the three  phases It is important to note that these predictions are independent 5Note that, unlike relations between the   , this sum rule is not RG invariant.Therefore, the numerical results presented in what follows are subject to corrections.These corrections can be readily computed in a given model [31].
of the value of .With these predictions, one can then determine the neutrinoless double beta decay matrix element, ⟨   ⟩, as shown in Figure 1.Given that only two out of the six conditions are utilized, the experimental best-fit values for the mixing parameters that have been used as our inputs in Figure 1 may not be fully consistent with all constraints (in fact they are not, as we will discuss below).Nevertheless, it is interesting to see even with one invariant, it is already possible to significantly constrain the model.Note that this analysis is independent of .The red-shaded region corresponds to the 3 disfavored range of values for the Dirac phase  from the global fit [32], while the gray-dashed line represents the current experimental upper bound for ⟨  ⟩ from the KamLAND-Zen collaboration [33].The projected sensitivities of future experiments such as nEXO [34] can reach values for ⟨   ⟩ ∼ 10 meV, and thus will probe the predictions of the invariant constraint  12 = −2 in this model.
After imposing Equation (28c) Equation (28a), there is only one observable left undetermined.If we impose equation (29), which entails two constraints, one for the real and one for the imaginary parts of  13  23 = −32, the system is overconstrained.We have verified that we cannot impose the constraints (28b) and (28c) while still being consistent with data.These findings are consistent with the analyses in [1,35], where it has been pointed out that one cannot accommodate all experimental data for the neutrino masses and mixings in this model.Note that we arrived at this conclusion without having to scan over .However, as discussed in [35], by adding one more parameter one can obtain a model which is remarkably consistent with the current experimental constraints.

A model based on 𝚪 5
We next consider a model based on Γ 5  5 [5,6,35].The assignments of modular weights and representations for the matter fields are shown in Table 2.This model introduces a singlet flavon, field/coupling     / (2) Table 2: Quantum numbers in the Γ 5 model.
, and an  5 triplet flavon, , which only couple to the charged leptons.The effective neutrino masses depend only on the modular forms of weight 2.More specifically, the relevant pieces of the superpotential are given by .

(33b)
The symmetries of the model forbid higher-order contributions to the effective neutrino mass operator in the superpotential.The flavons  and  are assumed to attain the VEVs With these VEVs, the charged lepton and neutrino mass matrices read where Here,  1,2,3,4,5 are the components of modular forms quintuplet (2) 5 ().They are not algebraically independent.In fact, each of them can be written as a homogeneous polynomial of 10 degrees in two basic modular forms of weight 1 /5,  1 () and  2 (),  [32]).
where [10,36] with  from (27), () being the Dedekind -function defined before, and the -constants given by The Hermitean combination  †    is diagonal, and the three charged lepton masses can be obtained by adjusting the free parameters , , , and .As before, we work in the basis in which the charged lepton Yukawa coupling is diagonal and the diagonal entries fulfill ( †    ) 11 < ( †    ) 22 < ( †    ) 33 .The best-fit value of modulus  is also close to the critical point i, These six real input parameters lead to the following neutrino mass and mixing parameters, as shown in Table 3. 6 The ratio of mass squared differences is given by While we have not found a set of input values that give rise to predictions that are consistent with all experimental data, it would still be interesting, as in the case of the Γ 3 Model, to see how 6Note that the precision with which we present predictions of the model is misleading in that we do not have sufficient theoretical control over the model.These are "mathematical predictions" which allow other research groups to cross-check our results.As discussed around Equation ( 5), there are limitations, and generally it is nontrivial to make the theoretical error bars smaller than the experimental ones, see [37] for a more detailed discussion.robust relations could arise by considering the invariants.The RG invariants emerging from the neutrino mass matrix (35b) are given by These RG invariants are meromorphic modular functions on Γ(5) rather than SL(2, Z).As a consequence, they can be written as rational polynomials of the Hauptmodul  5 () of Γ( 5).Further, the -expansions of Unlike in the Γ 3 model discussed in Section 4.1, none of the   is a constant.In particular,  12 has a singularity at  = i ∞,  13 is singular at  = 1+0.767664i 2 , and  23 is singular at  = 2+i 5 .Since  1,2,3,4,5 can be expressed in terms of two building blocks, cf.Equation ( 36), these three invariants   are also the rational polynomials of the same building blocks.
The way the algebraic relations between the invariants get specified is not unique.In what follows, we show one possible way, 0 = 4 + 18 These relations are richer than the corresponding constraint (24) in the Γ 3 model of Section 4.1.However, they have the same qualitative virtue as their pendants of Section 4.1: they allow us to derive constraints on the observables of the model.Interestingly, Equation ( 43) are invariant under the exchange  12 ↔  13 .At the level of the functions of observables (18), this transformation is equivalent to  23 ↦ →  23 + /2.This transformation is also known as  ↔  symmetry or 2 − 3 symmetry [38] (see e.g.[39] for a review), and has been considered in the context of modular flavor symmetries in [40].It is, therefore, worthwhile to explore the "fixed point" of this exchange symmetry, i.e. make the ansatz that 12 and  13 depend on  1 and  2 , and in the limit in which either of the   becomes zero at least one of the invariants becomes undefined.Therefore, we can assume  1 ≠ 0, and define  • • =  2 / 1 .
For instance, 9 = e 19 i/10 corresponds to  = −2/5 + i/5.Note that  = −2/5 + i/5 is a fixed point under the stabilizer Z 2 = {1,  2 ( 2 ) −1 }.At this fixed point, the neutrino mass matrix also has a generalized  ↔  symmetry, where and the −1 in Equation ( 47) comes from the automorphy factor ( + ) 2 = −1. (ii) We can express the RG invariants in terms of , Clearly, the   are rational functions of  5 .Since  5 is real,   are real.In fact, for all 10 solutions in (49), As one would expect,  23 is maximal, i.e. sin 2  23 = 1 /2, and sin 2  13 = 1 /5.The size of the mass eigenvalues depends only on whether  is even or odd, i.e. | While there are no exact analytic solutions, the following  values 5 +  , where  (II) with 0 < ,  ′ ≪ 1, solve (52) almost perfectly.The appearance of the relative phases between  (II)  can be seen easily from the definition of the   in (37).Note also that all  (II) 2 are related via SL(2, Z) but not Γ 5 transformations, and likewise for  (II) 2+1 .However, the solutions in (49) also predict the unrealistic relation  1 =  2 along with the sum rule  3 =  1 +  2 (and thus NO) and, as a consequence of ( 50), vanishing phases.The sum rule implies a continuous symmetry of the neutrino mass matrix, where () PMNS with  3 () being a rotation in the 1 − 2 plane.Furthermore, the predicted relations for the masses and mixing angles are a consequence of an approximate discrete symmetry of the neutrino mass matrix where the Hermitean unitary matrix  3 () squares to unity and is given by This transformation can be regarded as a Z 2 transformation of the 5-plet, We emphasize that none of the symmetries (55) are exact symmetries of the action, they are symmetries of neutrino mass matrix at the fixed point of the syzygies (43).However, in this setup having symmetries of the neutrino mass matrix, as opposed to symmetries of the action, comes at a price: the modular forms become very large, their absolute values can exceed 100.In the context of bottom-up model building this can be acceptable because there is no a priori normalization of the modular forms, i.e. we can always multiply them with a small constant.It is to be noted that symmetries of mass matrices have been discussed in the literature.However, to the best of our knowledge, these symmetries have been imposed in a rather ad hoc fashion in the sense that there is no model realization for these previous examples.We speculate that the type of model construction considered in this Letter may provide a consistent framework from which symmetries of mass matrices can arise dynamically.We will investigate this aspect further in a subsequent work.

Discussion
As we have seen, in modular invariant models of flavor, it is possible to relate certain meromorphic modular invariant functions to physical observables.In some cases, such as (28a) and ( 29), one even obtains modular invariant holomorphic observables, where a combination of observables conspires to become an integer, independent of the renormalization scale.We have shown that useful information and phenomenological constraints can be extracted from these relations, which, due to RG invariance, can be directly, modulo the limitations discussed around Equation ( 5), applied to observables measured in experiments.It will also be interesting to apply our discussion to the quark sector, where invariants were obtained from different considerations [41,42].The fact that these observables conspire to be integers may be regarded as a hint towards a topological origin of these relations.In the effective theory approach, it is not obvious how to substantiate such speculations.However, it has been known long before modular invariance was used in bottom-up model building that the couplings in string compactifications are modular forms [43, cf. the discussion around Equation ( 19)].Earlier work [44][45][46] and more recent analyses [23,[47][48][49][50][51][52][53][54][55][56][57][58][59] explore the stringy origin of these couplings.It will be interesting to see whether the abovementioned integers, which can be directly related to experimental observation as we have shown, play a special role in stringy completions of the SM.It is tempting to speculate that this may provide us with a direct relation between experimental measurements and properties of the compact dimensions.
Obviously, this is not the first time in which holomorphy ( τ ) and modular invariance ( ) is used to make firm physical predictions.In particular, the celebrated Seiberg-Witten theory [60,61] makes use of these concepts to solve gauge theories with  = 2 SUSY.However, our discussion shows, in the framework of modular flavor symmetries, , τ , and in some instances ∞ govern certain combinations of real-world observables.As we discussed, these combinations are RG invariant to all orders within  = 1 SUSY, and their poles and zeros are RG invariant even without SUSY.Let us reiterate that these conclusions are generally valid only under the assumption that the Kähler potential attains its minimal form (5) at some scale.It will, therefore, be interesting to find alternatives to [25] allowing us to control the Kähler potential.Likewise, the discussion of modular invariant holomorphic observables for non-minimal Kähler potentials are left to future work.

Summary
We have pointed out that in modular invariant models of flavor, certain combinations of couplings give rise to modular invariant meromorphic and even holomorphic physical observables.These objects are highly constrained by their symmetries and properties, RG invariant, and, at the same time, composed solely of quantities that can be measured experimentally.They carry a lot of information, and allow us to draw immediate, important, and robust conclusions on the model without the need to perform scans of the parameter space.In addition, symmetry relations among the invariants exist for certain modular symmetries, as illustrated in the Γ 5 model studies in this Letter.Fundamentally they are symmetries of the fixed points and can correspond to phenomenologically relevant ones, such as the  −  symmetry.
More importantly, to the best of our knowledge, these are the first examples in which physical observables are given by modular invariant functions.This Letter is only the start of exploiting their properties to obtain better theoretical control of model predictions.
where no summation over ,  is implied.It has been pointed out in [26] that certain ratios of entries of  do not depend on the renormalization scale, In the MSSM, this can be understood from the non-renormalization theorem.Here, only the wavefunction renormalization constants are scale-dependent, and this dependence precisely cancels in the above expressions [66].It has been noted in [26] that this statement also applies to the non-supersymmetric SM at the one-loop level.
The scale invariance of   is due to the fact that the renormalizable couplings in the SM have a larger global symmetry.Specifically, the lepton sector has global lepton family number symmetries.Therefore, in the basis in which the charged lepton mass matrix is diagonal, corrections that multiply the effective neutrino mass operator will be diagonal as well.As a result, where ,  and  are composed of the renormalizable couplings of the theory and diagonal, This has two immediate consequences: 1.At 1-loop, where   = 1 for all ,   are RG invariant.
2. Zeros and poles of   remain zeros and poles at all orders.
In particular, in the basis in which  is diagonal one can write the scale-dependent neutrino mass operator as as long as  does not have zeros.As indicated, only the   = √   are subject to RG evolution.In slightly more detail, only the absolute values of the   depend on the scale while their phases remain invariant.If one or more entries of  are zeros, then our discussion shows that these entries remain zero at all scales in the perturbative effective field theory (EFT) description.Zeros of the diagonal (off-diagonal) entries of  correspond to zeros (poles) of the   .This leads to RG invariant relations between the physical parameters, which will be studied elsewhere.

B More details on Feruglio Model
The purpose of this appendix is to show that the observables in Feruglio's Model 1 (cf.Section 4.1) are modular invariant, provided one transforms the VEV of   appropriately.To see this, recall that the superpotential terms in this model are given by contractions between the flavon   and the triplet of modular forms.The invariance of superpotential terms requires  4 invariance and that the modular weights of fields and modular forms involved in an operator to add up to zero.As for the former, the fact that The fields, including the flavon   , transform in such a way that the superpotential is invariant.The VEV of the flavon   is given by ⟨  ⟩ = (, 0, 0).Both this VEV and the invariants   () from ( 23) are invariant under  but not  transformations.However, the transformations which change   () can be regarded as a basis change, and after undoing the basis change the invariants get mapped to their original form.
In slightly more detail, under an  transformation Under the  transformation alone,   () from ( 23) are not invariant.However, once we diagonalize the charged lepton Yukawa couplings, which amounts to undoing (67b),   () get mapped back to their original form.Of course, these findings are a simple consequence of two basic facts: (i) modular transformations of () amount to transforming () with an  4 matrix and multiplying it by an automorphy factor, and (ii) invariants   are constructed in such a way that the automorphy factors cancel.Therefore, undoing the  4 transformation returns   to their original form.

)
Hence we can use the invariants to predict the values of the  phases in this model.Equation (28a) trivially fulfills requirements , τ & ∞ .It entails two constraints, Re  12 = −2 and Im  12 = 0. Therefore, for a given value of the Dirac  phase , we can predict the values of the Majorana phases  1 and  2 .

Figure 1 :
Figure 1: This figure displays the correlation between the neutrinoless double -decay matrix element ⟨   ⟩ and the  phase  for inverted ordering (IO) in the Γ 3 model, considering only the invariant constraint  12 = −2.Note that this analysis is independent of .The red-shaded region corresponds to the 3 disfavored range of values for the Dirac phase  from the global fit[32], while the gray-dashed line represents the current experimental upper bound for ⟨  ⟩ from the KamLAND-Zen collaboration[33].The projected sensitivities of future experiments such as nEXO[34] can reach values for ⟨   ⟩ ∼ 10 meV, and thus will probe the predictions of the invariant constraint  12 = −2 in this model.