# Explicit Perturbations to the Stabilizer τ= {\rm i} of Modular A^\prime_5 Symmetry and Leptonic CP Violation

EExplicit Perturbations to the Stabilizer τ = i of Modular A (cid:48) Symmetry and Leptonic CP Violation

Xin Wang a, b ∗ , Shun Zhou a, b † a Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, China b School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China

Abstract

In a class of neutrino mass models with modular ﬂavor symmetries, it has been observedthat CP symmetry is preserved at the ﬁxed point (or stabilizer) of the modulus parameter τ = i, whereas signiﬁcant CP violation emerges within the neighbourhood of this stabilizer.In this paper, we ﬁrst construct a viable model with the modular A (cid:48) symmetry, and explorethe phenomenological implications for lepton masses and ﬂavor mixing. Then, we introduceexplicit perturbations to the stabilizer at τ = i, and present both numerical and analyticalresults to understand why a small deviation from the stabilizer leads to large CP violation.As low-energy observables are very sensitive to the perturbations to model parameters, wefurther demonstrate that the renormalization-group running eﬀects play an important role inconfronting theoretical predictions at the high-energy scale with experimental measurementsat the low-energy scale. ∗ E-mail: [email protected] † E-mail: [email protected] (corresponding author) a r X i v : . [ h e p - ph ] F e b Introduction

The experimental discovery of neutrino oscillations indicates that neutrinos are actually massiveand leptonic ﬂavor mixing is signiﬁcant, but both the origin of neutrino masses and the leptonicﬂavor structure are largely unknown at present [1, 2]. One of the simplest ways to accommodatetiny neutrino masses is to extend the standard model (SM) by introducing three right-handedneutrino singlets N i R (for i = 1 , ,

3) and implementing the canonical seesaw mechanism [3–7],which attributes the smallness of three ordinary neutrino masses to the largeness of three heavyMajorana neutrino masses. On the other hand, the leptonic ﬂavor mixing can be accounted forby imposing some discrete ﬂavor symmetries on the seesaw model. In these models, a numberof scalar ﬁelds transforming nontrivially under the ﬂavor symmetry group have to be introduced,and the ﬂavor structures of lepton mass matrices depend much on the assignments of the relevantﬁelds into the representations of the symmetry group and the choices of the vacuum expectationvalues (vev’s) of the scalar ﬁelds [8–11].Recently, the modular symmetry has been suggested as a possible solution to the ﬂavor mixingpuzzle [12]. In this framework, the Yukawa couplings in the leptonic sector turn out to be modularforms, which are holomorphic functions of the complex modulus τ and transform as irreduciblerepresentations of ﬁnite modular groups Γ N (with N ≥ τ acquires its vev, the Yukawa couplings are determined and thus leptonic ﬂavor mixingpattern is obtained. Finite modular groups, such as Γ (cid:39) S [13–15], Γ (cid:39) A [16–27], Γ (cid:39) S [28–33], Γ (cid:39) A [34–36], Γ (cid:39) PSL(2 , Z ) [37], and some of their double coverings Γ (cid:48) (cid:39) A (cid:48) [38, 39], Γ (cid:48) (cid:39) S (cid:48) [40, 41] and Γ (cid:48) (cid:39) A (cid:48) [42, 43], have been extensively investigated in theprevious literature.In the bottom-up approach to the modular-invariant ﬂavor models, the modulus τ is usuallytreated as a free parameter, which can be pinned down by ﬁtting it together with other modelparameters to experimental observations. In contrast, in the top-down approach, it can be ﬁxedby the modulus stabilization via the minimum of the supergravity scalar potential [44] or the ﬂuxcompactiﬁcations [45]. Interestingly, residual symmetries have been found after the spontaneousbreaking of the global modular symmetry at some special values of τ , which are called ﬁxed pointsor stabilizers [46, 47]. If τ is exactly located at one of the stabilizers, it seems unlikely to realizeviable lepton ﬂavor models by using only one modular symmetry [48, 49] or a common value of τ in both the charged-lepton and neutrino sectors [46,50]. One can alternatively construct modular-invariant models with the modulus close to the stabilizers, with which the strong hierarchy ofcharged-lepton masses can be successfully realized [51, 52].As pointed out in Ref. [53], if the generalized CP (gCP) symmetry is further imposed onthe modular-invariant model, then both CP and modular symmetries are spontaneously brokenby the vev of τ . Moreover, all the stabilizers are CP conserving, while a small deviation of themodulus from them may result in large CP violation [53]. Motivated by these observations, wepropose in this paper a feasible lepton ﬂavor model with the gCP symmetry and the Γ (cid:48) (cid:39) A (cid:48) symmetry. A salient feature of our model is the prediction for a nearly-maximal CP violationwhen τ ≈ i holds, whereas CP symmetry is preserved at τ = i. It is worth mentioning that amodular-invariant model with large CP violation around the stabilizer τ = i of the modular A τ = i remains to be answered. To this end, starting with themodel at the stabilizer τ = i, we introduce explicit perturbations to the modulus and other modelparameters, and derive the approximate analytical formulas of neutrino masses, mixing anglesand CP-violating phases. Through these analytical calculations, we can see that the nearly-maximal CP violation mainly comes from the large ratio between the real and imaginary parts of τ . In addition, theoretical predictions for some low-energy observables are very sensitive to theperturbations to model parameters when τ is approaching i. Such ﬁndings also imply that therenormalization-group (RG) running eﬀects may have remarkable impact on the ﬂavor parameters.Therefore, we also take into account the RG running eﬀects and verify that this is indeed the case.The remaining part of this paper is structured as follows. In Sec. 2 we give a brief introductionto the modular symmetry and the modular Γ (cid:48) (cid:39) A (cid:48) group in order to establish our notations.Then the concrete model for lepton masses and ﬂavor mixing based on the modular A (cid:48) symmetry isconstructed in Sec. 3. The phenomenological implications for low-energy observables are studiedin Sec. 4 in a completely numerical way, while the analytical analysis of the perturbations tothe model with the stabilizer τ = i is performed in Sec. 5. Radiative corrections to the mixingparameters via the RG running are discussed in Sec. 6. We summarize our main results in Sec. 7.Finally, the basic properties of the ﬁnite modular group Γ (cid:48) (cid:39) A (cid:48) are presented in Appendix A. In this section, we brieﬂy review some basic knowledge about modular symmetries and the modularΓ (cid:48) (cid:39) A (cid:48) group in order to establish our notations and set up the framework for later discussions.The modular group is isomorphic to the special linear group SL(2 , Z ), which is deﬁned as [12]Γ ≡ (cid:40)(cid:32) a bc d (cid:33) (cid:12)(cid:12)(cid:12)(cid:12) a, b, c, d ∈ Z , ad − bc = 1 (cid:41) . (2.1)There are in total three generators S , T and R for the modular group Γ, the matrix representationsof which are given by S = (cid:32) − (cid:33) , T = (cid:32) (cid:33) , R = (cid:32) − − (cid:33) . (2.2)For the modular-invariant supersymmetric theories, the action S should be unchanged under themodular transformations on the modulus τ and the chiral supermultiplet χ ( I ) , namely, γ : τ → aτ + bcτ + d , χ ( I ) → ( cτ + d ) − k I ρ ( I ) ( γ ) χ ( I ) , (2.3)where γ is an element of the modular group Γ, k I is the weight of the chiral supermultiplet and ρ ( I ) ( γ ) denotes the representation matrix of γ . From Eq. (2.3) one can easily observe that thetransformations on τ induced by γ and − γ are actually the same, while the matter ﬁelds χ ( I ) are generally allowed to transform nontrivially under R . Therefore, one should consider Γ ratherthan Γ ≡ Γ / { I , − I } as the symmetry group in such theories. In this case, we can introduce the3ouble covering of ﬁnite modular groups Γ (cid:48) N ≡ Γ / Γ( N ) with Γ( N ) being the principal congruencesubgroups of the modular group Γ, e.g., Γ (cid:48) (cid:39) A (cid:48) , Γ (cid:48) (cid:39) S (cid:48) and Γ (cid:48) (cid:39) A (cid:48) .The modular form f ( τ ) of level N and weight k is a holomorphic function of τ , which transformsunder Γ( N ) as f ( γτ ) = ( cτ + d ) k f ( τ ) , γ ∈ Γ( N ) , (2.4)where k ≥ M k [Γ( N )],the modular forms can always be decomposed into several multiplets that transform as irreducibleunitary representations of Γ (cid:48) N . To be more precise, we can always ﬁnd a proper basis of the modularspace M k [Γ( N )] such that a modular multiplet Y ( k ) r = ( f ( τ ) , f ( τ ) , · · · ) T in the representation r satisﬁes the following equation Y ( k ) r ( γτ ) = ( cτ + d ) k ρ r ( γ ) Y ( k ) r ( τ ) , γ ∈ Γ (cid:48) N , (2.5)where ρ r ( γ ) denotes the representation matrix of γ .For Γ (cid:48) (cid:39) A (cid:48) in question, the group structure and modular forms with weights from one tosix have been analyzed in detail in Ref. [42]. Hence we just recapitulate the key points relevantfor the following discussions. The A (cid:48) group has 120 elements, which can be produced by threegenerators S , T and R satisfying the identities S = R , ( ST ) = I , T = I , R = I , RT = T R . (2.6)There are nine distinct irreducible representations for A (cid:48) , where the representations , , (cid:48) , and with R = I coincide with those for A , whereas (cid:98) , (cid:98) (cid:48) , (cid:98) and (cid:98) are unique for A (cid:48) with R = − I .The irreducible representation matrices of S , T and R , as well as the decomposition rules of theKronecker products relevant for the present work, can be found in Appendix A.The modular forms Y (1) (cid:98) ( τ ) with the lowest nontrivial weight k = 1 can be expressed as thelinear combinations of six basis vectors (cid:98) e i (for i = 1 , · · · ,

6) in the modular space M [Γ(5)], whoseexplicit forms as well as Fourier expansions are given in Appendix A. Six components of Y (1) (cid:98) ( τ )will be denoted as Y i ( τ ) (for i = 1 , , · · · , Y (1) (cid:98) = Y Y Y Y Y Y = (cid:98) e − (cid:98) e √ (cid:98) e (cid:98) e (cid:98) e √ (cid:98) e − (cid:98) e − (cid:98) e , (2.7)where the argument τ of all relevant functions has been suppressed. In addition, we write downthe modular forms that will be used for the model building in Sec. 3. For weight two, we have Y (2) (cid:48) = 12 √ Y − Y Y − Y ) −√ Y ( Y + Y ) √ Y ( Y − Y ) , Y (2) = 12 √ Y + Y )2 √ Y (2 Y + Y ) √ Y ( Y − Y ) √ Y ( Y + 3 Y )2 √ Y (2 Y − Y ) . (2.8)4or weight three, we will use Y (3) (cid:98) , = − √ Y − Y Y − Y Y − Y − Y ( Y − Y Y − Y ) Y ( Y + Y ) ( Y + 2 Y ) Y ( Y − Y ) (2 Y − Y )2 Y (2 Y − Y Y − Y )2 Y − Y Y + 12 Y Y + 5 Y ,Y (3) (cid:98) , = − √ Y − Y Y − Y Y + Y − Y (2 Y − Y Y − Y )2 Y Y Y Y Y ( Y − Y Y − Y ) − Y − Y Y + 9 Y Y + 3 Y . (2.9)For weight four, two modular forms Y (4) and Y (4) are involved, namely, Y (4) = −√ (cid:0) Y − Y Y − Y Y + 3 Y Y + Y (cid:1) ,Y (4) = √ ( Y + Y ) (7 Y − Y Y − Y ) Y (13 Y − Y Y − Y Y − Y ) − Y (9 Y − Y Y + 3 Y Y + 13 Y ) . (2.10)As we have mentioned, once the modulus τ acquires its vev, the modular symmetry will bespontaneously broken down. However, there are some special values of τ , which are stabilizers [29]and keep unchanged under the transformations induced by one or more generators of the modulargroup. Consequently, the global modular symmetry is only partially broken, and some residualsymmetries are left in the theory. Consider the fundamental domain G of the modular group Γ,which is deﬁned as G = { τ ∈ C : Im τ > , − . ≤ Re τ ≤ . , | τ | ≥ } . (2.11)There are four kinds of stabilizers in this fundamental domain: • τ C = i, which is invariant under S ; • τ L = − / √ /

2, invariant under ST ; • τ R = +1 / √ /

2, invariant under

T S ; • τ T = i ∞ , invariant under T .In the present paper, we concentrate on the stabilizer τ C = i, which is invariant under thetransformation corresponding to S , i.e., τ → − /τ . Furthermore, it is straightforward to verifythat the residual symmetry at τ = i is Z S = { I , S, R, RS } , where I stands for the identity element.5 Modular A (cid:48) Model

Now we are ready to construct a concrete model based on the modular A (cid:48) symmetry. To beginwith, we need to assign properly the weights and irreducible representations to all the superﬁeldsunder the modular group A (cid:48) . Diﬀerent from most of the previous models, the superﬁelds ofleft-handed lepton doublets (cid:98) L will not be arranged as a triplet of A (cid:48) in our model. Instead, wetake (cid:98) L e ∼ and (cid:98) L µτ ≡ ( (cid:98) L µ , (cid:98) L τ ) T ∼ (cid:98) . Correspondingly, the superﬁelds of right-handed chargedleptons are also put into a singlet and a doublet of A (cid:48) , i.e., (cid:98) E C1 ∼ and (cid:98) E C23 ≡ ( (cid:98) E C2 , (cid:98) E C3 ) T ∼ (cid:98) .Since the two-dimensional modular forms exist only with the odd weights, the ﬂavor structureof the charged-lepton mass matrix M l could be highly constrained. As we shall show soon, M l in our model is restricted to be block-diagonal. In the neutrino sector, the superﬁelds of threeright-handed neutrinos N C are set to be (cid:48) of A (cid:48) . In addition, two Higgs doublets (cid:98) H u and (cid:98) H d are both assumed to be in the trivial one-dimensional irreducible representation. All these chargeassignments of the superﬁelds in our model are listed in Table 1.With the representations and weights of the supermultiplets in Table 1, one can write downthe superpotentials relevant for lepton masses W l = ξ (cid:104)(cid:98) L e (cid:98) E C1 (cid:105) (cid:98) H d + ξ (cid:104)(cid:16)(cid:98) L µτ (cid:98) E C23 (cid:17) Y (4) (cid:105) (cid:98) H d + ξ (cid:104)(cid:16)(cid:98) L µτ (cid:98) E C23 (cid:17) Y (4) (cid:105) (cid:98) H d , W D = g (cid:20)(cid:16)(cid:98) L e (cid:98) N C (cid:17) (cid:48) Y (2) (cid:48) (cid:21) (cid:98) H u + g (cid:104)(cid:16)(cid:98) L µτ (cid:98) N C (cid:17) (cid:98) Y (3) (cid:98) , (cid:105) (cid:98) H u + g (cid:104)(cid:16)(cid:98) L µτ (cid:98) N C (cid:17) (cid:98) Y (3) (cid:98) , (cid:105) (cid:98) H u , W R = 12 Λ (cid:104)(cid:16) (cid:98) N C (cid:98) N C (cid:17) Y (2) (cid:105) . (3.1)Implementing the Kronecker product rules for A (cid:48) collected in Appendix A, we arrive at the explicitform of the charged-lepton mass matrix M l = v d √ ξ √ ξ (cid:16) Y (4) (cid:17) ξ (cid:34) √ (cid:16) Y (4) (cid:17) − √ (cid:101) ξY (4) (cid:35) ξ (cid:34) √ (cid:16) Y (4) (cid:17) + √ (cid:101) ξY (4) (cid:35) − √ ξ (cid:16) Y (4) (cid:17) ∗ , (3.2)with ξ /ξ ≡ (cid:101) ξ , the Dirac neutrino mass matrix M D = √ g v u (cid:16) Y (2) (cid:48) (cid:17) (cid:16) Y (2) (cid:48) (cid:17) (cid:16) Y (2) (cid:48) (cid:17) + (cid:101) g −√ (cid:16) Y (3) (cid:98) , (cid:17) −√ (cid:16) Y (3) (cid:98) , (cid:17) (cid:16) Y (3) (cid:98) , (cid:17) − (cid:16) Y (3) (cid:98) , (cid:17) −√ (cid:16) Y (3) (cid:98) , (cid:17) (cid:16) Y (3) (cid:98) , (cid:17) + (cid:16) Y (3) (cid:98) , (cid:17) −√ (cid:16) Y (3) (cid:98) , (cid:17) + (cid:101) g −√ (cid:16) Y (3) (cid:98) , (cid:17) −√ (cid:16) Y (3) (cid:98) , (cid:17) (cid:16) Y (3) (cid:98) , (cid:17) − (cid:16) Y (3) (cid:98) , (cid:17) −√ (cid:16) Y (3) (cid:98) , (cid:17) (cid:16) Y (3) (cid:98) , (cid:17) + (cid:16) Y (3) (cid:98) , (cid:17) −√ (cid:16) Y (3) (cid:98) , (cid:17) ∗ , (3.3)6able 1: The charge assignment of the chiral superﬁelds under the SU(2) L gauge symmetry andthe modular A (cid:48) symmetry in our model, with the corresponding weights listed in the last row. (cid:98) L e (cid:98) L µτ (cid:98) E C1 (cid:98) E C23 (cid:98) N C (cid:98) H u , (cid:98) H d SU(2) 2 2 1 1 1 2 A (cid:48) (cid:98) (cid:98) (cid:48) − k I − g /g ≡ (cid:101) g and g /g ≡ (cid:101) g , and the Majorana mass matrix of right-handed neutrinos M R = Λ4 (cid:16) Y (2) (cid:17) −√ (cid:16) Y (2) (cid:17) −√ (cid:16) Y (2) (cid:17) −√ (cid:16) Y (2) (cid:17) √ (cid:16) Y (2) (cid:17) − (cid:16) Y (2) (cid:17) −√ (cid:16) Y (2) (cid:17) − (cid:16) Y (2) (cid:17) √ (cid:16) Y (2) (cid:17) ∗ , (3.4)where ( Y ( k ) r ) i denotes the i -th element in the multiplet Y ( k ) r . Given M D and M R , we can immedi-ately derive the eﬀective neutrino mass matrix from the seesaw formula, i.e., M ν ≈ − M D M − M TD .Hence the lepton mass spectra and ﬂavor mixing parameters can be extracted from the charged-lepton mass matrix M l and the eﬀective neutrino mass matrix M ν .Apart from the modulus τ , the lepton mass matrices involve three extra parameters (cid:101) ξ , (cid:101) g and (cid:101) g , which are in general complex. All these complex parameters may contribute to leptonicCP violation. Following Ref. [53], we further impose gCP symmetry on our modular A (cid:48) modelsuch that the modulus τ becomes the only source of CP violation. More explicitly, the gCPtransformation of the superﬁeld χ ( I ) ( x ) is χ ( I ) ( x ) CP −→ X r χ ( I ) ( x P ) , (3.5)where χ ( I ) ( x P ) denotes the conjugate superﬁeld with x P = ( t, − (cid:126)x ) and X r represents a unitarymatrix acting on the ﬂavor space. For the gCP symmetry to be consistent with the modularsymmetry, the following condition should be satisﬁed X r ρ ∗ r ( γ ) X − r = ρ r ( u ( γ )) , (3.6)where u ( γ ) is an outer automorphism of the modular group. The consistency condition given inEq. (3.6) indicates that the modulus τ transforms under CP as τ CP −→ − τ ∗ . (3.7) It has been mentioned in Ref. [40] that there are two distinct kinds of outer automorphisms u ( γ ) for the doublecovering group, corresponding to two diﬀerent gCP symmetries denoted as CP and CP , respectively. However, inthe basis where the representation matrices of S and T are both symmetric, X r will always be the trivial identitymatrix, i.e., X r = I , regardless of whether CP or CP is combined with the Γ (cid:48) group. σ and 3 σ intervals, together with the values of σ i being thesymmetrized 1 σ uncertainties, for three neutrino mixing angles { θ , θ , θ } , two neutrino mass-squared diﬀerences { ∆ m , ∆ m or ∆ m } and the Dirac CP-violating phase δ from a global-ﬁtanalysis of current experimental data [56].Parameter Best ﬁt 1 σ range 3 σ range σ i Normal neutrino mass ordering ( m < m < m )sin θ .

304 0.292 — 0.317 0.269 — 0.343 0.0125sin θ . θ .

570 0.546 — 0.588 0.407 — 0.618 0.021 δ/ ◦

195 170 — 246 107 — 403 38∆ m / (10 − eV ) 7 .

42 7.22 — 7.63 6.82 — 8.04 0.205∆ m / (10 − eV ) +2 .

514 +2.487 — +2.542 +2.431 — +2.598 0.0275Inverted neutrino mass ordering ( m < m < m )sin θ .

304 0.292 — 0.317 0.269 — 0.343 0.0125sin θ . θ .

575 0.554 — 0.592 0.411 — 0.621 0.019 δ/ ◦

286 254 — 313 192 — 360 29.5∆ m / (10 − eV ) 7 .

42 7.22 — 7.63 6.82 — 8.04 0.205∆ m / (10 − eV ) − . − .

525 — − . − .

583 — − .

412 0 . X r = I with I being the identity element if the representation matrices of both S and T are symmetric [53]. Furthermore, as can be seen in Appendix A, all the Clebsch-Gordan coeﬃcientsare real, so the modular forms Y ( k ) r will transform under CP as Y ( k ) r ( τ ) CP −→ Y ( k ) r ( − τ ∗ ) = (cid:2) Y ( k ) r ( τ ) (cid:3) ∗ . (3.8)Therefore, to render the superpotentials invariant under the gCP transformation, we require all thecoupling constants in our model to be real. As a result, the whole symmetry of modular and gCPtransformations is spontaneously broken down after the modulus τ gets its vev. However, thereare some special values of τ , for which CP symmetry is conserved while the modular symmetryis broken. As pointed out in Ref. [53], these values are located along the imaginary axis (i.e.,Re τ = 0) and the boundary of the fundamental domain G . Thanks to the gCP symmetry, only eight real parameters { Re τ, Im τ, ξ v d , ξ v d , (cid:101) ξ , g v / Λ , (cid:101) g , (cid:101) g } are involved in our model. The allowed regions of model parameters can be found by followingthe numerical analysis adopted in Ref. [42], and the basic strategy is summarized as below. • First, we explain the experimental results that are used to constrain the parameter space. Forthe charged-lepton masses, we take the best-ﬁt values m e = 0 .

510 MeV, m µ = 107 . m τ = 1 .

840 GeV, which are evaluated at the scale Λ

GUT ≈ × GeV of granduniﬁed theories (GUT) with tan β ≡ v u /v d = 10 and the supersymmetry breaking scale m SUSY = 10 TeV in Ref. [55]. With the help of these charged-lepton masses, one candetermine the model parameters ξ v d , ξ v d and (cid:101) ξ if the complex modulus τ is given. Forthe neutrino sector, we take the best-ﬁt values, as well as 1 σ and 3 σ ranges, of two neutrinomass-squared diﬀerences ∆ m ≡ m − m and ∆ m ≡ m − m in the normal mass ordering(NO) with m < m < m or ∆ m and ∆ m ≡ m − m in the inverted mass ordering (IO)with m < m < m , three ﬂavor mixing angles { θ , θ , θ } , and the Dirac CP-violatingphase δ , from the global-ﬁt analysis by NuFIT 5.0 [56,57] without including the atmosphericneutrino data from Super-Kamiokande. All these values are summarized in Table 2. • The model predictions for low-energy observables, including charged-lepton and neutrinomasses, ﬂavor mixing angles, and CP-violating phases, can be obtained by diagonalizinglepton mass matrices M l and M ν . Then, the compatibility between the model predictionsand the experimental observations is measured by the χ -function, which is constructed asthe sum of several one-dimensional functions χ j , namely, χ ( p i ) = (cid:88) j χ j ( p i ) , (4.1)where p i ∈ { Re τ, Im τ, (cid:101) g , (cid:101) g } stand for the model parameters, and j is summed over theobservables { sin θ , sin θ , sin θ , r } with r ≡ ∆ m / ∆ m (∆ m / | ∆ m | ) in the NO(IO) case. Here we do not include the information of δ in the χ -function due to the weakconstraints on δ from the global-ﬁt results. For sin θ , sin θ and r , we make use of theGaussian approximations χ j ( p i ) = (cid:32) q j ( p i ) − q bf j σ j (cid:33) , (4.2)where q j ( p i ) denote the model predictions for these observables and q bf j are their best-ﬁtvalues from the global analysis in Ref. [56]. The associated uncertainties σ j are derived bysymmetrizing 1 σ uncertainties from the global-ﬁt analysis, as given in Table 2. For sin θ ,we utilize the one-dimensional projection of the χ -function provided by Refs. [56, 57]. Byminimizing the overall χ -function in Eq. (4.1), we can determine the best-ﬁt values of themodel parameters { Re τ, Im τ, (cid:101) g , (cid:101) g } . The remaining overall factor g v / Λ in the neutrinosector then can also be ﬁxed if one of the neutrino mass-squared diﬀerences is given.After carrying out the numerical analysis, we ﬁnd that our model can be compatible withthe experimental data at the 3 σ level only in the NO case. In Fig. 1, the 3 σ allowed parameterspace of model parameters as well as the predictions for low-energy observables are shown. Somecomments on the numerical results are in order.1. In the top-left panel of Fig. 1, we can observe that the 3 σ allowed value of Re τ is larger than − .

16, while the upper bound of Im τ is around 1 .

03. In fact, there should be a duplicateregion of { Re τ, Im τ } in the right-half part of the fundamental domain G with 0 ≤ Re τ ≤ . ●● ●● ●● ● ●● ● ●● ● ●●●● ● ●●● ●● ● ●●● ●● ●● ●●●●● ● ●● ●● ●●●● ● ●●●●●● ●●●● ●● ●●●● ●●●● ● ● ●● ● ●● ●●● ● ● ●● ●●●●● ●●●● ●● ●●● ●● ●● ●●● ● ●● ●●●●● ●●● ●●● ●● ●● ●● ●●●●● ●●● ● ● ● ●●● ●●● ● ●● ● ●● ●● ●● ●● ●●● ●● ●●●● ● ●●● ● ●● ●● ●● ● ●●●● ● ●● ●● ●●● ● ●● ●●● ●●● ●●●● ●● ●● ●● ●● ●●● ●●● ●●● ●●● ● ●● ● ●●●●●●● ●●● ●● ●● ●● ●●● ●● ● ●● ●● ●●●● ●●● ● ●●● ● ●● ●●● ● ●● ● ●● ●●● ● ●●●●● ●●●●● ● ● ● ●●●●● ● ● ●●● ● ● ● ●●● ●● ●● ●●● ●● ● ●●●● ●● ●● ●● ● ●●●● ● ●●●●●● ● ●● ●●● ●● ● ●● ●● ●● ●● ● ●● ● ●● ●●● ●●● ●● ●● ●●●●● ●● ●●● ● ● ●●● ●●● ● ●● ●● ●●● ●●●●●● ● ●●● ●● ●● ● ●● ●●● ●●● ● ●● ●●● ●●● ●●● ●● ● ●●● ●●● ●● ●●● ●●●● ●●● ●● ●●● ●● ●● ●● ●●● ●●● ●●● ●● ●●●● ● ●●● ●● ●● ●●●●● ●● ● ●● ●● ●● ● ●● ●● ●● ●●● ●● ● ●●●● ●● ● ●●● ●●● ●● ● ●● ●●●●●● ●●● ●● ●●● ● ●● ●● ●● ● ●●● ●●● ● ●●●● ● ●●● ● ●● ●● ●●●● ●●●● ● ●● ●● ●●● ●●●●● ●● ●●● ● ● ●●● ● ●● ●●● 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●●● ● ●● ●●● ●●●● ●●● ●● ●●● ●● ● ★★ - - - - τ Σ m ν / e V ★★ - - m / eV m ββ / e V KamLAND-Zen Upper Bound

Figure 1: The allowed parameter space of the model parameters { Re τ, Im τ, (cid:101) g , (cid:101) g } and low-energyobservables at the 3 σ level in the NO case, where the cyan stars denote the best-ﬁt values fromthe χ -ﬁt analysis. The gray shaded region in the bottom-right panel represents the upper boundon m ββ from the KamLAND-Zen experiment [58], and the blue boundary is obtained by using the3 σ ranges of { θ , θ } and { ∆ m , ∆ m } from the global-ﬁt analysis.where the signs of all CP-violating phases are reversed. However, the predicted value of theDirac CP-violating phase δ in this region is constrained to be about 90 ◦ , which is lyingoutside the 3 σ allowed range of δ from the global-ﬁt analysis. The allowed regions of theother two parameters { (cid:101) g , (cid:101) g } in the neutrino sector are shown in the top-right panel ofFig. 1, where it is worthwhile to mention that the relation (cid:101) g = − (cid:101) g / (cid:101) g . Some of the low-energy observables are strongly correlatedwith Re τ (or Im τ ). For example, the middle-left panel of Fig. 1 reveals that the value of θ increases as Re τ decreases. When the value of Re τ is approaching zero, θ becomesvery close to 45 ◦ . Interestingly, the Dirac CP-violating phase δ shows a similar behavior,i.e., δ tends to 270 ◦ when Re τ →

0, implying a nearly-maximal CP violation in the vicinityof τ = i. This result seems quite confusing, since the stabilizer τ = i should lead to CPconservation. Why does a small deviation from τ = i (i.e., Im τ − < . θ and θ and two neutrino mass-squared diﬀerences are loosely constrained andthus not presented here.2. The sum of three light neutrino masses (cid:80) m ν = m + m + m also depends crucially on thevalue of Re τ . As indicated in the bottom-left panel of Fig. 1, the minimal value of (cid:80) m ν pre-dicted in our model is 0 .

15 eV, which runs into contradiction with the most stringent bound (cid:80) m ν < .

12 eV from the Planck observations of cosmic microwave background [59]. Noticethat the limit (cid:80) m ν < .

12 eV has also been gained earlier in Ref. [60, 61]. However, thisupper bound is dependent on the choices of observational data sets and cosmological models.The latest global-ﬁt analysis of absolute neutrino masses yields (cid:80) m ν < (0 . · · · .

69) eVat the 2 σ level [62], where the “aggressive” and “conservative” combinations of diﬀerentobservational data sets are considered. Therefore, our model is still consistent with the“conservative” cosmological bound on neutrino masses.3. Two Majorana CP-violating phases ρ and σ can also be determined, so we can calculate theeﬀective mass for neutrinoless double-beta decays m ββ ≡ | m cos θ cos θ e ρ + m cos θ sin θ e σ + m sin θ e − δ | , (4.3)where the standard parametrization of leptonic ﬂavor mixing matrix has been taken [63].The numerical result of m ββ is presented in the bottom-right panel of Fig. 1, where wecan ﬁnd that the predicted range of m ββ is overlapping with the upper bound m upper ββ =(61 · · · m ββ ∼

15 meV [64] will be ableto make a ﬁnal verdict on whether our model is ruled out or not.Based on the χ -ﬁt analysis, we ﬁnd that the minimum χ = 0 .

716 is reached in the NO casewith the following best-ﬁt values of the model parametersRe τ = − . , Im τ = 1 . , (cid:101) g = 1 . , (cid:101) g = − . . (4.4)Combining the above values with charged-lepton masses, we obtain ξ v d = 0 .

721 MeV, ξ v d =0 .

247 GeV and (cid:101) ξ = 3 . g v / Λ = 63 .

05 meV.These best-ﬁt values of model parameters lead to a nearly-degenerate spectrum of three neutrinomasses m = 73 .

27 meV, m = 73 .

78 meV, m = 89 .

10 meV, and three mixing angles θ = 33 . ◦ , θ = 8 . ◦ and θ = 42 . ◦ . Meanwhile, the predictions for three CP-violating phases are11 = 271 . ◦ , ρ = 88 . ◦ and σ = 90 . ◦ . In addition, the eﬀective mass for neutrinoless double-beta decays is m ββ = 71 .

78 meV. All these predictions are readily to be tested in future neutrinooscillation and neutrinoless double-beta decay experiments.

As we have seen in Sec. 4, our model predicts a nearly-maximal CP-violating phase δ = 271 . ◦ ,given Re τ = − . τ = 1 . τ = i. Since CP viola-tion is completely absent for τ = i, it deserves a further investigation to clarify how such signiﬁcantCP violation is generated. In this section, we attempt to explore the analytical properties of ourmodel in the parameter space, where τ is close to the stabilizer τ = i, by introducing explicitperturbations to the stabilizer and calculating lepton mass spectra and ﬂavor mixing parameters.Let us ﬁrst consider the NO case. If τ = i exactly holds, with the help of the explicit expressionsof basis vectors (cid:98) e i (for i = 1 , · · · ,

6) shown in Eq. (A.4), we can ﬁnd (cid:98) e (i) (cid:98) e (i) = (cid:98) e (i) (cid:98) e (i) = (cid:98) e (i) (cid:98) e (i) = (cid:98) e (i) (cid:98) e (i) = (cid:98) e (i) (cid:98) e (i) = A , (5.1)where A = (cid:112) √ φ − φ with φ ≡ ( √ /

2. The elements of Y (1) (cid:98) (i) can be explicitly written as Y (i) = (cid:98) e (i)(1 − A ) , Y (i) = 5 √ (cid:98) e (i) A , Y (i) = 10 (cid:98) e (i) A ,Y (i) = 10 (cid:98) e (i) A , Y (i) = 5 √ (cid:98) e (i) A , Y (i) = − (cid:98) e (i)(3 + A ) . (5.2)Then the modular forms at τ = i with higher weights can be constructed by using Eq. (5.2).Since the stabilizer τ = i keeps unchanged under the transformation of S , H l ≡ M l M † l from thecharged-lepton sector is invariant under S , i.e., H l = ρ † l ( S ) H l ρ l ( S ) , (5.3)where ρ l ( S ) ≡ Diag { , ρ ( S ) } with ρ ( S ) being the representation matrix of the two-dimensionalirreducible representation of S in the A (cid:48) group. The identity in Eq. (5.3) tells us that a unitarymatrix converting ρ l ( S ) into its diagonal form can also diagonalize the matrix H l . It is easy toverify that both ρ l ( S ) and H l can be diagonalized by the real orthogonal matrix U l = θ l − sin θ l θ l cos θ l , (5.4)where tan 2 θ l = (cid:112) ( φ − /φ . Two comments on U l in Eq. (5.4) are helpful. First, substituting φ = ( √ / ≈ .

618 into Eq. (5.4), we arrive at cos θ l ≈ .

962 and sin θ l ≈ . U l is roughly an identity matrix, and its contribution to the lepton ﬂavor mixing is insigniﬁcant.Second, we have checked that the form of U l given in Eq. (5.4) holds as a good approximation tothe unitary matrix that diagonalizes H l , even if τ slightly deviates from the stabilizer τ = i. It isthen safe to assume U l in Eq. (5.4) to be valid in the vicinity of τ = i.12e turn to the neutrino sector. Without loss of generality, one can work in the ﬂavor basiswhere the charged-lepton mass matrix is diagonal, and redeﬁne the eﬀective neutrino mass matrixas (cid:102) M ν = U † l M ν U l . For τ = i, (cid:102) M ν is simply written as (cid:102) M ν = (cid:98) g v Det( M R ) .

67 0 − . (cid:98) g + 4 (cid:98) g )0 0 0 − . (cid:98) g + 4 (cid:98) g ) 0 5 . (cid:98) g + 4 (cid:98) g ) , (5.5)with (cid:98) g = g | (cid:98) e ( τ ) | and (cid:98) g , = (cid:101) g , | (cid:98) e ( τ ) | . It is obvious that the right-handed neutrino massmatrix M R has one zero eigenvalue for τ = i, i.e., Det( M R ) = 0 in Eq. (5.5). The seesaw formula M ν = − M D M − M TD is not directly applicable for τ = i, but anyway we are interested in the regionwhere τ = i + (cid:15) with | (cid:15) | (cid:28)

1. In this case, the determinant of M R is found to be Det( M R ) ∝ iΛ (cid:15) to the ﬁrst order of (cid:15) . Although (cid:15) is a small parameter, the overall factor Λ corresponding to themass scale of right-handed neutrinos can be quite large, giving rise to a sizable value of Det( M R ).On the other hand, Eq. (5.5) indicates that if (cid:98) g = − (cid:98) g /

4, only the (1,1)-element of (cid:102) M ν survives.As can be seen from the top-right panel of Fig. 1, the ratio (cid:98) g / (cid:98) g = (cid:101) g / (cid:101) g is indeed around − (cid:101) g / (cid:101) g = − .

25 that is not far from the best-ﬁt value (cid:101) g / (cid:101) g ≈ − .

28. Inspiredby this observation, we further assume (cid:98) g = − (cid:98) g / τ = i, and introduceperturbations to this identity when τ deviates from i.In the following discussions, we consider explicit perturbations to both τ = i and (cid:98) g = − (cid:98) g / • First, we introduce the perturbation to the stabilizer, i.e., τ = i + (cid:15) , where (cid:15) ≡ (cid:15) R + i (cid:15) I is acomplex parameter with | (cid:15) | = (cid:112) (cid:15) + (cid:15) (cid:28) (cid:15) R ( (cid:15) I ) being the real (imaginary) part. Inthe presence of this perturbation, the ratios of two adjacent basis vectors of M [Γ(5)] read (cid:98) e (i + (cid:15) ) (cid:98) e (i + (cid:15) ) = (cid:98) e (i + (cid:15) ) (cid:98) e (i + (cid:15) ) = (cid:98) e (i + (cid:15) ) (cid:98) e (i + (cid:15) ) = (cid:98) e (i + (cid:15) ) (cid:98) e (i + (cid:15) ) = (cid:98) e (i + (cid:15) ) (cid:98) e (i + (cid:15) ) ≈ A (1 + 1 .

245 i (cid:15) ) , (5.6)to the ﬁrst order of (cid:15) . With Eq. (5.6), we can also write down the approximate expressionsof all modular forms for τ = i + (cid:15) . Then the neutrino mass matrix (cid:102) M ν up to O ( (cid:15) ) is (cid:102) M ν ≈ − (cid:98) g v Λ (cid:15) .

089 i − . (cid:15) − .

823 i (cid:98) g (cid:15) (cid:98) g (cid:15) (1 .

258 + 5 .

731 i (cid:15) )0 (cid:98) g (cid:15) (1 .

258 + 5 .

731 i (cid:15) ) 2 .

332 i (cid:98) g (cid:15) ∗ , (5.7)where the identity (cid:98) g = − (cid:98) g / (cid:15) R < (cid:15) I >

0. It is then evident that H ν ≡ (cid:102) M ν (cid:102) M † ν can be diagonalized via U † H ν U = Diag { m , , m , , m , } by the following unitary matrix U = − i · θ − sin θ θ cos θ , (5.8)where sin θ ≈ √ / . (cid:15) R . This simple formula implies that θ is approximately 45 ◦ when Re τ = (cid:15) R is small in magnitude, which is in good agreement with numerical results13n the middle-left panel of Fig. 1. In addition, three eigenvalues of H ν can be expressed as m , ≈ (0 . − . (cid:15) I − . (cid:15) ) µ ,m , ≈ (1 .

583 + 3 . (cid:15) R − . (cid:15) I + 35 . (cid:15) ) (cid:98) g (cid:15) µ ,m , ≈ (1 . − . (cid:15) R − . (cid:15) I + 35 . (cid:15) ) (cid:98) g (cid:15) µ , (5.9)where µ ≡ ( (cid:98) g v ) / (Λ | (cid:15) | ). Note that we have retained the terms up to next-to-leading orderof (cid:15) R but only the leading order terms of (cid:15) I . This treatment has been guided by the numericalresults, which demonstrate that | (cid:15) R | is about ten times larger than (cid:15) I in the allowed parameterspace. Eq. (5.9) indicates that the values of m , and m , are proportional to (cid:98) g (cid:15) , whichshould be highly suppressed because of a small value of (cid:15) R . However, the coeﬃcients in theparentheses on the right-hand side of Eq. (5.9) for m , and m , can be much larger thanthat for m , . Therefore, it is possible to have the feasible parameter space of (cid:98) g where thenormal mass ordering is allowed. • From previous discussions, we have observed that the perturbation to τ = i gives θ ≈ ◦ ,and the identity (cid:98) g = − (cid:98) g / θ and θ . In order to accommodaterealistic mixing angles, we have to break the identity by assuming (cid:98) g = − (cid:98) g / κ , where κ is another perturbative parameter. To the ﬁrst order of κ , the matrix H ν after the (2 , H (cid:48) ν = (cid:98) H ν + ∆ H ν with (cid:98) H ν = Diag { m , , m , , m , } , where theleading-order mass eigenvalues have been given in Eq. (5.9), and the perturbation matrix∆ H ν is given by∆ H ν ≈ µ κ . − i +i+i 0 0 − i 0 0 + 0 . (cid:98) g − i (cid:15) ∗ − i (cid:15) +i (cid:15) (cid:15) ∗ . (5.10)One can numerically check that the (1 , H ν is about ten times larger than its(1 , (cid:15) and (cid:98) g are within their individual 3 σ allowed range. Therefore, weimplement a sequence of rotations, namely, the (1 , , H (cid:48) ν to diagonalize it. For the (1 , U approximates to U = e i ϕ · cos θ − sin θ θ θ , (5.11)where sin θ ≈ µ | κ | (0 . − . (cid:98) g (cid:15) R ) m , − m , , ϕ ≈ arctan (cid:18) . − (cid:98) g (cid:15) R (cid:98) g (cid:15) I (cid:19) . (5.12)From Eq. (5.12) we see that θ is proportional to the new perturbative parameter κ , so itis always possible to get sin θ ∼ .

15 by adjusting properly the value of κ . It is apparentthat ∆ H ν also induces some corrections to the eigenvalues m , and m , , which turn out tobe proportional to κ and thus can be neglected. After the above (1 , σ allowed regions of { Re τ, Im τ } and { (cid:101) g , (cid:101) g } obtained by inputting the approx-imate analytical formulas of three neutrino masses and mixing angles are represented by blueshaded areas. For comparison, the allowed regions obtained by exact numerical calculations areplotted in red.to H (cid:48) ν , we have H (cid:48)(cid:48) ν = U † H (cid:48) ν U ≈ m , − . µ (cid:98) g (cid:15) I κe − i ϕ − . µ (cid:98) g (cid:15) I κe i ϕ m ,

00 0 m , , (5.13)where the approximation cos θ ≈ . µ (cid:98) g (cid:15) I κ is much smaller than m , and m , in magnitude, but thehigh degeneracy between m , and m , ensures a relatively large mixing angle θ . By usingthe degenerate perturbation theory, we obtain m ≈ m , − . µ (cid:98) g (cid:15) I | κ | ,m ≈ m , + 0 . µ (cid:98) g (cid:15) I | κ | ,m ≈ m , , (5.14)which are the ﬁnal results for three light neutrino masses. Meanwhile, the unitary matrix U to diagonalize H (cid:48)(cid:48) ν is found to be U = e − i ϕ · cos θ − sin θ θ cos θ

00 0 1 , (5.15)with sin θ ≈ . µ (cid:98) g (cid:15) I | κ | m , − m , + 0 . µ (cid:98) g (cid:15) I | κ | , (5.16)15 in δ ρ σ Figure 3: The distributions of { sin δ, sin ρ, sin σ } in the vicinity of τ = i, where (cid:101) g = 1 . (cid:101) g = − .

350 are ﬁxed. The vertical dotted line corresponds to Re τ = − . τ in its 3 σ allowed parameter space.implying that the value of sin θ can be greatly enhanced if m , is very close to m , . Forexample, if m , − m , ≈ . µ (cid:98) g (cid:15) I | κ | or equivalently ∆ m ≈ . µ (cid:98) g (cid:15) I | κ | , we willobtain sin θ ≈ /

3, which agrees well with the experimental observation. To illustratehow well the analytical approximations are in agreement with exact numerical results, wehave also scanned over the model parameters by using the approximate expressions of threeneutrino masses and mixing angles. The 3 σ allowed regions of { Re τ, Im τ, (cid:101) g , (cid:101) g } are shownin Fig. 2, where an excellent agreement with the exact numerical results can be found for | Re τ | (cid:46) . (cid:101) g (cid:38) . U , U and U together, we ﬁnally obtain the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix [65, 66] U = U U U = c c s c s e i ϕ − s c − c s s e − i ϕ c c − s s s e − i ϕ s c s s − c c s e − i ϕ − c s − s c s e − i ϕ c c , (5.17)where s ij ≡ sin θ ij and c ij ≡ cos θ ij (for ij = 12 , ,

23) have been deﬁned, and the unphysicalphases have been eliminated by redeﬁning the phases of the charged-lepton ﬁelds. ComparingEq. (5.17) with the standard parametrization of U [63], we can extract the Dirac CP-violatingphase δ as δ = 2 π − ϕ ≈ π − arctan (cid:18) . − (cid:98) g (cid:15) R (cid:98) g (cid:15) I (cid:19) . (5.18)Since the value of | (cid:15) R | is much larger than (cid:15) I in the allowed parameter space, we obtain a nearly-maximal Dirac CP-violating phase, i.e., δ ≈ ◦ . Two Majorana CP-violating phases can beﬁgured out by diagonalizing the neutrino mass matrix via U † (cid:102) M ν U ∗ = Diag { m e i ρ , m e i σ , m } . Itturns out that ρ ≈ π (cid:18) (cid:15) I (cid:15) R (cid:19) , σ ≈ π − arctan (cid:18) (cid:15) I (cid:15) R (cid:19) , (5.19)which are both close to 90 ◦ since (cid:15) I (cid:28) | (cid:15) R | . In the left panel of Fig. 3, we present the distributionof sin δ in the vicinity of τ = i where (cid:101) g = 1 .

456 and (cid:101) g = − .

350 are ﬁxed for illustration. For16 | - - - - - - Figure 4: The distribution of the absolute value of Jarlskog invariant |J | in the vicinity of τ = i,where (cid:101) g = 1 .

456 and (cid:101) g = − .

350 are ﬁxed. The black star corresponds to Re τ = − . τ = 1 . { Re τ, Im τ } in their 3 σ allowed parameter space.the chosen values of (cid:101) g and (cid:101) g , the 3 σ allowed value of Re τ is tightly restricted to be around − . δ goes down as Im τ increases, which can be well described by Eq. (5.18). For completeness,we also show the distributions of sin ρ and sin σ around τ = i in the right two panels of Fig. 3.Note that the values of sin ρ and sin σ along some parts of the imaginary axis as well as thelower boundary of the fundamental domain G can be maximal, corresponding to ρ = σ = 90 ◦ .However, CP violation associated with Majorana phases should be always proportional to sin 2 ρ or sin 2 σ , implying CP conservation along the imaginary axis and the lower boundary of G .One can also adopt a parametrization-independent approach to measure the magnitude of CPviolation. For instance, the Jarlskog invariant J ≡ Im (cid:2) U e U ∗ e U ∗ µ U µ (cid:3) for CP violation in leptonicsector [67, 68] can be calculated via I = − m ∆ m ∆ m ∆ m eµ ∆ m µτ ∆ m τe J , (5.20)where ∆ m αβ ≡ m α − m β (for α, β = e, µ, τ ) denote the mass-squared diﬀerences of chargedleptons and I ≡ Tr (cid:8) [ H ν , H l ] (cid:9) is one of the CP-odd weak-basis invariants [69–72]. Substitutingthe approximate expressions of H ν and H l into I and using Eq. (5.20), we obtain J ≈ . µ κ (cid:98) g (cid:15) ( (cid:15) I − . (cid:15) )∆ m ∆ m ∆ m , (5.21)where one can observe that J may be highly suppressed due to a tiny numerator on the right-handside. Therefore, a sizable value of J can be achieved if ∆ m is small enough. For example, if wetake ∆ m = 0 . µ (cid:98) g (cid:15) I | κ | , corresponding to sin θ = 1 / J ≈ . r (cid:15) ( (cid:15) I − . (cid:15) ) | κ | (cid:15) (1 − r ) , (5.22)where the relation r = ∆ m / ∆ m in the NO case has been used. The maximal CP violation with δ ≈ ◦ corresponds to J = s c s c s c sin δ ≈ − .

03, if all the mixing angles take theirindividual best-ﬁt values from the global-ﬁt analysis as shown in Table 2. Assuming (cid:15) R ≈ − . (cid:15) I ≈ .

003 in Eq. (5.22), we can ﬁnd that

J ≈ − .

03 requires κ ≈ r ≈ . |J | in the vicinity of τ = i with (cid:101) g = 1 .

456 and (cid:101) g = − .

350 is shown in Fig. 4,where one can clearly see that the red ring of radius (cid:112) (cid:15) + (cid:15) ∼ .

035 corresponds to the relativelylarge values of J . The 3 σ allowed values of { Re τ, Im τ } denoted by the black star are exactlylocated at this ring, indicating the nearly-maximal CP violation.Before closing this section, let us brieﬂy discuss why the IO case is not permitted in our model.For the IO case, we can also introduce explicit perturbations to τ = i in a similar way. After theﬁrst (2 , , H (cid:48) ν becomes much larger than the counterpart in the NOcase. As a consequence, if we require sin θ ∼ / r ≡ ∆ m / | ∆ m | will be close to one, whichis apparently incompatible with the experimental observation. The modular symmetry and seesaw mechanism are supposed to work at a very high energy scaleΛ SS , whereas the ﬂavor mixing parameters are measured at the electroweak scale characterizedby the mass m Z ≈ . Z gauge boson. In order to confront theoretical predictionswith experimental observations, one has to take into account the radiative corrections to physicalparameters via their RG equations. It is well known that the RG running eﬀects can be signiﬁ-cant, especially for large values of tan β and (or) nearly-degenerate neutrino masses [73]. Similarconclusions have also been reached in the modular-invariant ﬂavor models [32,74]. As indicated inEq. (5.16), the value of sin θ in our model is very sensitive to small changes of model parameters,particularly in the region close to τ = i. Hence it is reasonable to expect large corrections fromRG running even if the value of tan β is relatively small.For deﬁniteness, we ﬁrst carry out a quantitative analysis of RG running eﬀects on the mixingparameters from the GUT scale Λ SS ∼ Λ GUT to the electroweak scale m Z , where tan β = 10 willbe assumed. In the basis where the charged-lepton Yukawa coupling matrix (cid:101) Y l ≡ Diag { y e , y µ , y τ } is diagonal with y α = √ m α /v d (for α = e, µ, τ ), the Dirac neutrino Yukawa coupling matrixbecomes (cid:101) Y ν = U † l Y ν with Y ν = √ M D /v u . After integrating out heavy Majorana neutrinos belowthe scale Λ SS , we obtain the eﬀective neutrino mass parameter M ≡ − U † l Y ν M − Y T ν U ∗ l . In theminimal supersymmetric SM, the radiative corrections to (cid:101) Y l and M can be described by theone-loop RG equations [75–80]16 π d (cid:101) Y l d t = (cid:104) α l + 3 (cid:16) (cid:101) Y l (cid:101) Y † l (cid:17)(cid:105) (cid:101) Y l , (6.1)16 π d M d t = α ν M + (cid:20)(cid:16) (cid:101) Y l (cid:101) Y † l (cid:17) M + M (cid:16) (cid:101) Y l (cid:101) Y † l (cid:17) T (cid:21) , (6.2)18igure 5: The allowed regions of low-energy observables { r, θ , θ , δ } versus the model parameterRe τ without (with) taking account of the corrections from RG running eﬀects are denoted by thered (blue) shaded areas.where t ≡ ln( µ/ Λ SS ) with µ being the renormalization scale, α l = − . g − g + 3Tr( Y d Y † d ) +Tr( (cid:101) Y l (cid:101) Y † l ) and α ν = − . g − g + 6Tr( Y u Y † u ). Notice that g and g represent respectivelythe SU(2) L and U(1) Y gauge couplings, Y u and Y d denote the up- and down-type quark Yukawacoupling matrices. In the chosen ﬂavor basis where (cid:101) Y l is diagonal, only the diagonal elements onboth sides of Eq. (6.1) survive. Therefore, the radiative corrections to neutrino masses and leptonﬂavor mixing parameters are determined by the RG equation of M [81]. The solution to M inEq. (6.2) can be formally written as M ( m Z ) = I ν I e I µ

00 0 I τ M (Λ SS ) I e I µ

00 0 I τ , (6.3)19here I ν and I α (for α = e, µ, τ ) are the evolution functions [81] I ν = exp (cid:34) − π (cid:90) ln(Λ SS /m Z )0 α ν ( t )d t (cid:35) ,I α = exp (cid:34) − π (cid:90) ln(Λ SS /m Z )0 y α ( t )d t (cid:35) . (6.4)While I ν in Eq. (6.3) aﬀects only the absolute scale of light neutrino masses, I α could modify bothneutrino masses and ﬂavor mixing parameters. By using Eq. (6.3), one can establish the directconnection between the eﬀective neutrino mass parameter at the high-energy scale Λ SS and thatat the electroweak scale m Z .To illustrate radiative corrections to ﬂavor mixing parameters, we numerically solve the RGequations and search for the allowed parameter space by re-scanning over the model parameters.It turns out that the 3 σ allowed regions of { Re τ, Im τ, (cid:101) g , (cid:101) g } obtained with or without radiativecorrections are hardly distinguishable, implying that the RG running eﬀects will not change theallowed parameter space. However, they can greatly modify the predictions for ﬂavor mixingparameters. To make this point clearer, we identify the 3 σ allowed parameter space in Sec. 4 as theinput at the GUT scale, and calculate the predictions for neutrino masses and mixing parametersat the electroweak scale with radiative corrections. In Fig. 5, we present the correlations betweenlow-energy observables { r, θ , θ , δ } and the model parameter Re τ without and with RG runningcorrections. Notice that the correction to θ is negligibly small, so we do not show the result of θ . As one can see from Fig. 5, the RG running eﬀects become larger for smaller values of | Re τ | .In particular, when | Re τ | is as small as 0 .

03, the mixing angle θ can reach 55 ◦ , which agreeswell with the previous result from explicit perturbations.To better understand why radiative corrections to θ are so large, we consider the RG runningeﬀects on the model parameters. Since the electron and muon Yukawa couplings are extremelysmall, I e ≈ I µ ≈ I τ contributes dominantly to therunning eﬀects on M in Eq. (6.3). With the help of Eq. (6.3), we can derive the approximateformula of (cid:102) M ν ( m Z ) around the stabilizer τ = i after including the RG running. Then, three lightneutrino masses at the electroweak scale are m ( m Z ) ≈ m , − . I τ µ (cid:98) g (cid:15) I | κ | ,m ( m Z ) ≈ I τ m , + 0 . I τ µ (cid:98) g (cid:15) I | κ | ,m ( m Z ) ≈ I τ m , , (6.5)and the expression of sin θ ( m Z ) is approximately given bysin θ ( m Z ) ≈ . I τ µ (cid:98) g (cid:15) I | κ | I τ m , − m , + 0 . I τ µ (cid:98) g (cid:15) I | κ | , (6.6)which will be reduced to Eq. (5.16) for I τ = 1. As pointed out in the last section, m , canbe very close to m , when Re τ approaches zero. Therefore, although I τ − ≈ − . I τ m , − m , in the denominator on the right-hand side of Eq. (6.6)leads to large corrections to sin θ . For instance, let us set the values of model parameters20o be (cid:15) = − . . (cid:98) g = 1 .

555 and κ = 0 . θ ≈ . ◦ .On the other hand, if the radiative corrections are taken into consideration, Eq. (6.6) will give θ ( m Z ) ≈ . ◦ , where one can observe a remarkable modiﬁcation of θ due to the RG runningeﬀects. In conclusion, despite the fact that the RG running eﬀects on the model parameters areinconspicuous, the low-energy observables are sensitive to small perturbations to free parametersin the vicinity of τ = i and radiative corrections are actually playing a signiﬁcant role.Although our analytical calculations are performed in a speciﬁc model, they are indeed help-ful in understanding sizable radiative corrections to the mixing angle θ for nearly-degenerateneutrino masses in the most general case. The key point is that the large mixing angle θ isessentially determined by the ratio of two small model parameters, to which radiative correctionsthemselves are tiny but change considerably the ratio. Modular ﬂavor symmetries provide us with an attractive way to account for lepton ﬂavor mixingand CP violation. Instead of just ﬁtting the modulus τ to the experimental data, one can alsostart with one of some special values τ (i.e., the ﬁxed points or stabilizers), at which residualsymmetries are retained after the breaking of the global modular symmetry. Any realistic modelsfor lepton masses, ﬂavor mixing and CP violation require the modulus τ to slightly deviate from thestabilizers. Therefore, it is interesting to explore the basic properties of modular-symmetry modelswith a modulus τ in the vicinity of the stabilizer. Such an exploration is useful for understandingthe phenomenological implications of the modular-symmetry models, and gives a clue to possibleresidual symmetries and the modular symmetry itself.In this paper, we construct a feasible lepton ﬂavor model based on the modular A (cid:48) groupcombined with the gCP symmetry, and focus on the ﬁxed point τ = i where CP symmetry ispreserved. A seemingly puzzling feature is that it predicts a nearly-maximal CP-violating phasein the region where τ = i + (cid:15) with (cid:15) being a small parameter. By treating (cid:15) as a perturbationto the stabilizer τ = i, we derive the approximate analytical expressions of light neutrino masses,ﬂavor mixing angles and the CP-violating phases. We ﬁnd that sin θ depends on the mass-squared diﬀerence ∆ m , and the enhancement of sin θ can be attributed to the high degeneracybetween two neutrino masses m and m . The approximate expression of tan δ is found to be − (0 . − (cid:98) g (cid:15) R ) / ( (cid:98) g (cid:15) I ), where (cid:15) R ≡ Re (cid:15) is about ten times larger than (cid:15) I ≡ Im (cid:15) in the 3 σ allowed parameter space. This analytical result shows that a nearly-maximal CP-violating phasecan be obtained near τ = i. We ﬁnd that (cid:15) ≈ − .

03 + 0 .

003 i is suﬃcient to give

J ≈ − . δ ≈ ◦ when all the mixing angles areconsistent with their observed values.Since sin θ is very sensitive to small perturbations to model parameters in the region around τ = i, radiative corrections via RG running to the model parameters should be important. Giventhe same input parameters and Re τ ∼ − .

03, the mixing angle θ would be changed from 32 ◦ to 50 ◦ after the radiative corrections are taken into consideration. The main reason for such asigniﬁcant RG running eﬀect is that the mass degeneracy between m and m is governed bysmall perturbation parameters while θ is determined by the ratio of two small perturbation21arameters. Although the details of such an analysis in this paper are quite model-dependent,it does shed some light on the common features of the modular-symmetry models that seem tobe “unstable” around the stabilizers. For instance, neutrino masses are usually predicted to benearly degenerate in such kinds of models.In the near future, it is necessary and interesting to further investigate the properties ofmodular-symmetry models around all the stabilizers in a systematic and model-independent way.As such scenarios are strictly constrained by the modular symmetry and the residual symmetries,their predictions for neutrino masses, mixing angles and the Dirac CP-violating phase are readilyto be tested in next-generation neutrino oscillation experiments, neutrinoless double-beta decaysand cosmological observations. Acknowledgements

This work was supported in part by the National Natural Science Foundation of China undergrant No. 11775232 and No. 11835013, and by the CAS Center for Excellence in Particle Physics.22

Modular A (cid:48) group and the modular space M [Γ(5)] The A (cid:48) group has 120 elements, which can be divided into nine conjugacy classes, indicating that A (cid:48) has nine distinct irreducible representations that are normally denoted as , (cid:98) , (cid:98) (cid:48) , , (cid:48) , , (cid:98) , and (cid:98) by their dimensions. The representation matrices of all three generators S , T and R inthe irreducible representations are summarized as below : ρ ( S ) = +1 , ρ ( T ) = +1 , ρ ( R ) = +1 , (cid:98) : ρ ( S ) = i √ (cid:32) √ φ √ φ − √ φ − −√ φ (cid:33) , ρ ( T ) = (cid:32) ω ω (cid:33) , ρ ( R ) = − I × , (cid:98) (cid:48) : ρ ( S ) = i √ (cid:32) √ φ − √ φ √ φ −√ φ − (cid:33) , ρ ( T ) = (cid:32) ω ω (cid:33) , ρ ( R ) = − I × , : ρ ( S ) = 1 √ −√ −√ −√ − φ φ − −√ φ − − φ , ρ ( T ) = ω

00 0 ω , ρ ( R ) = + I × , (cid:48) : ρ ( S ) = 1 √ − √ √ √ − φ φ √ φ − φ , ρ ( T ) = ω

00 0 ω , ρ ( R ) = + I × , : ρ ( S ) = 1 √ φ − φ − φ − − φφ − φ − − φ φ − , ρ ( T ) = ω ω ω

00 0 0 ω , ρ ( R ) = + I × , (cid:98) : ρ ( S ) = i5 −√ φ + 1 √ φ (cid:112) φ − √ φ − √ φ √ φ − √ φ + 1 (cid:112) φ − (cid:112) φ − √ φ + 1 −√ φ − −√ φ √ φ − (cid:112) φ − −√ φ √ φ + 1 ,ρ ( T ) = ω ω ω

00 0 0 ω , ρ ( R ) = − I × , : ρ ( S ) = 15 − √ √ √ √ √ φ − − φ φ − φ √ − φ φ ( φ − φ − √ φ −

1) ( φ − φ − φ √ φ φ − − φ ( φ − ,ρ ( T ) = ω ω ω

00 0 0 0 ω , ρ ( R ) = + I × , : ρ ( S ) = − i5 √ φ − (cid:112) φ − √ φ − −√ φ + 1 √ φ √ φ − − (cid:112) φ − −√ φ − (cid:112) φ − √ φ √ φ √ φ √ φ − (cid:112) φ − √ φ −√ φ − −√ φ √ φ + 1 −√ φ + 1 √ φ −√ φ − −√ φ (cid:112) φ − √ φ − √ φ √ φ −√ φ (cid:112) φ − √ φ − (cid:112) φ − √ φ − √ φ √ φ + 1 √ φ − (cid:112) φ − −√ φ ,ρ ( T ) = ω ω ω ω

00 0 0 0 0 1 , ρ ( R ) = − I × , where ω ≡ e π/ and I n × n denotes the n -dimensional identity matrix. Notice that the representa-tions , , (cid:48) , and with R = I coincide with those for A , whereas (cid:98) , (cid:98) (cid:48) , (cid:98) and (cid:98) are uniquefor A (cid:48) with R = − I .The whole list of decomposition rules for the Kronecker products of any two nontrivial irre-ducible representations of A (cid:48) can be found in Ref. [42]. Here we only summarize the decompositionrules relevant to this paper. • (cid:98) ⊗ (cid:98) = a ⊕ s a : − √

22 ( α β − α β ) , s : √ α β + α β −√ α β √ α β . • (cid:98) ⊗ (cid:48) = (cid:98) (cid:98) : − √ α β − α β √ α β −√ α β √ α β −√ α β α β + α β . • ⊗ = s ⊕ a ⊕ s s : √

33 ( α β + α β + α β ) , a : √ α β − α β α β − α β α β − α β , s : √ α β − α β − α β −√ α β − √ α β √ α β √ α β −√ α β + α β ) . (cid:48) ⊗ (cid:48) = s ⊕ (cid:48) a ⊕ s s : √

33 ( α β + α β + α β ) , (cid:48) a : √ α β − α β α β − α β α β − α β , s : √ α β − α β − α β √ α β −√ α β + α β ) −√ α β + α β ) √ α β . • ⊗ = s ⊕ a ⊕ (cid:48) a ⊕ s ⊕ a ⊕ s , ⊕ s , s : √

55 ( α β + α β + α β + α β + α β ) , a : √ 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) 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 , s : √ √ α β + √ α β − α β + 4 α β − α β √ α β + 4 α β + √ α β − α β − α β √ α β − α β − α β + √ α β + 4 α β √ α β − α β + 4 α β − α β + √ α β , 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 , s , : √ 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 , s , : √ 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 . • (cid:98) ⊗ (cid:98) = a ⊕ s , ⊕ s , ⊕ (cid:48) s , ⊕ (cid:48) s , ⊕ s ⊕ a ⊕ s ⊕ a , ⊕ a , a : √

66 ( α β + α β − α β + α β − α β − α β ) , s , : 12 α β + α β + α β + α β α β + α β + α β + α β α β − α β − α β + α β , s , : √ − 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) s , : √ 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) s , : √ √ 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 , s : √ − α β − α β − α β + α β − √ α β + α β − α β √ α β + 2 √ α β + √ α β + 2 √ α β + α β + α β + 2 √ α β √ α β − α β − α β + 2 √ α β − √ α β + 2 √ α β − √ α β α β + α β − √ α β + α β + α β − α β − α β , 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 − α β + α β , s : √ √ α β + α β ) √ (cid:0) α β − α β − √ α β − α β + α β (cid:1) − 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:0) α β + α β + √ α β + α β + α β (cid:1) . a , : √ 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 ) , a , : √ √ α β + α β − α β − α β ) − α β + 2 α β + α β + α β − α β − α β −√ (cid:0) α β − α β − √ α β + √ α β (cid:1) √ (cid:0) √ α β − √ α β − α β + α β (cid:1) α β − α β + α β − α β + 2 α β − α β . For a given non-negative integer k , the modular space M k [Γ(5)] of weight k for Γ(5) contains5 k + 1 linearly-independent modular forms, which can be regarded as the basis vectors of themodular space. According to Ref. [82], we have M k [Γ(5)] = (cid:77) a + b =5 ka,b ≥ C η (5 τ ) k η ( τ ) k k a , (5 τ ) k b , (5 τ ) , (A.1)where η ( τ ) is the Dedekind eta function η ( τ ) = q / ∞ (cid:89) n =1 (1 − q n ) , (A.2)26ith q ≡ e πτ , and k r ,r ( τ ) is the Klein form k r ,r ( τ ) = q ( r − / z (1 − q z ) × ∞ (cid:89) n =1 (1 − q n q z ) (cid:0) − q n q − z (cid:1) (1 − q n ) − , (A.3)with ( r , r ) being a pair of rational numbers in the domain of Q − Z , z ≡ τ r + r and q z ≡ e πz .Now we take k = 1, then the basis vectors of the modular space M [Γ(5)] can be expressed as (cid:98) e ( τ ) = η (5 τ ) η ( τ ) k , (5 τ ) , (cid:98) e ( τ ) = η (5 τ ) η ( τ ) k , (5 τ ) k , (5 τ ) , (cid:98) e ( τ ) = η (5 τ ) η ( τ ) k , (5 τ ) k , (5 τ ) , (cid:98) e ( τ ) = η (5 τ ) η ( τ ) k , (5 τ ) k , (5 τ ) , (cid:98) e ( τ ) = η (5 τ ) η ( τ ) k , (5 τ ) k , (5 τ ) , (cid:98) e ( τ ) = η (5 τ ) η ( τ ) k , (5 τ ) . (A.4)Furthermore, making use of Eqs. 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