Mu-tau reflection symmetry with a high scale texture-zero

The $\mu\tau$-reflection symmetric neutrino mass matrix can accommodate all known neutrino mixing angles, with maximal atmospheric angle fixed, and predicts all the unknown CP phases of the lepton sector but is unable to predict the absolute neutrino mass scale. Here we present a highly predictive scenario where $\mu\tau$-reflection is combined with a discrete abelian symmetry to enforce a texture-zero in the mass matrix of the heavy right-handed neutrinos that generate the light neutrino masses. Such a restriction reduces the free parameters of the low energy theory to zero and the absolute neutrino mass scale is restricted to few discrete regions, three in the few meV range and one extending up to around 30 meV. The heavy neutrino sector is dependent only on two free parameters which are further restricted to small regions from the requirement of successful leptogenesis. Mass degenerate heavy neutrinos are possible in one case but there is no resonant enhancement of the CP asymmetry.

The µτ -reflection symmetric neutrino mass matrix can accommodate all known neutrino mixing angles, with maximal atmospheric angle fixed, and predicts all the unknown CP phases of the lepton sector but is unable to predict the absolute neutrino mass scale. Here we present a highly predictive scenario where µτ -reflection is combined with a discrete abelian symmetry to enforce a texture-zero in the mass matrix of the heavy right-handed neutrinos that generate the light neutrino masses. Such a restriction reduces the free parameters of the low energy theory to zero and the absolute neutrino mass scale is restricted to few discrete regions, three in the few meV range and one extending up to around 30 meV. The heavy neutrino sector is dependent only on two free parameters which are further restricted to small regions from the requirement of successful leptogenesis. Mass degenerate heavy neutrinos are possible in one case but there is no resonant enhancement of the CP asymmetry.

I. INTRODUCTION
Our picture of the parameters that govern neutrinos physics at low energy are almost complete after the measurement of nonzero reactor angle in 2012 [1]. In case neutrinos are Dirac, only the absolute neutrino mass, the mass ordering and one Dirac CP phase is unknown. The measurement of this CP phase is one of the goals of current experimental efforts to advance our knowledge about neutrinos. In case neutrinos are Majorana, two more Majorana CP phases should be added to the list of unknowns.
One of the simplest symmetries that can predict all the CP phases and yet allow CP violation is the symmetry known as µτ -reflection symmetry or CP µτ symmetry where the neutrino sector is invariant by exchange of the muon neutrino with the tau antineutrino [2,3]; see also review in [4]. This symmetry predicts maximal Dirac CP phase (δ = ±90 • ) and trivial Majorana phases with discrete choices of the CP parities. Additionally, the atmospheric angle θ 23 is predicted to be maximal (45 • ), well within 1σ in the latest global fits [5]. The recent IceCube results on atmospheric neutrinos also corroborate maximal θ 23 [6]. Current data also hints at a value of the Dirac CP phase in the broad vicinity of −90 • . As a consequence of the symmetry, the fixed values for the CP phases lead to characteristic bands for the possible effective mass of neutrinoless double beta decay but still allows successful leptogenesis [7] to occur if flavor effects are taken into account [8]; see also Ref. [9] for a review on leptogenesis in the presence of flavor symmetries. If the conditions for maximal atmospheric angle and Dirac CP phase are relaxed, correlations between θ 23 and δ can be tested in the future DUNE and Hyper-K experiments [10]. Even the exact CP µτ case can be tested in DUNE [11] but CP µτ is too simple to predict the other unknown parameter, i.e, the absolute neutrino mass scale.
In that respect, it was shown in Ref. [12] that the imposition of an abelian discrete symmetry in conjunction with CP µτ symmetry could enforce a one-zero texture in addition to the CP µτ form. Such a setting reduced the number of free parameters in the neutrino mass matrix from five to four to account for the four observables ∆m 2 21 , ∆m 2 32 , θ 12 , θ 13 where cp denotes the usual CP conjugation. It was shown in Ref. [8] that Z 8 was the minimal abelian symmetry where a nontrivial combination with CP µτ is possible. We can think that these two symmetries -Z 8 and CP µτ -initially act on the left-handed lepton doublets (L e , L µ , L τ ) before they are spontaneously broken. Then the two symmetries act on the same space and CP µτ induces the following automorphism on Z 8 [8]: where T encodes the Z 8 transformation in (2) and X denotes ν µ -ν τ interchange in (3) represented by We also note that the rephasing transformations that preserve Z 8 in (2) and CP µτ in (3) are of the form It is clear that these transformations also preserve the form of the mass matrix in (1) and can be used to make c or d real. Flavor independent rephasing by i also preserves the form of the mass matrix (flips the sign of a, b) but changes CP µτ by a global sign. Hence, only the relative sign of a and b is significant.
In the charged lepton sector, the µτ mass difference arises from a large source of CP µτ breaking at high energy [8]; see appendix A for more details. After that stage, the Z 8 will remain as a residual symmetry so that we are simply left with We assume that the physics responsible for such a CP µτ breaking is well above the scale of the heavy neutrinos which come from Z 8 breaking.
Light neutrino masses will come from the type I seesaw mechanism where we add three singlet neutrinos N αR , α = e, µ, τ . The N αR and left-handed lepton doublets L α transform under Z 8 and CP µτ in the same way as in eqs. (2) and (3). So the neutrino Dirac mass matrix will be diagonal.
To avoid bare terms, we also introduce a Z B−L 4 symmetry under which the lepton doublets L α and the singlet neutrinos N αR carry charge −i. Heavy neutrino masses will be generated by singlet scalars η k with Z B−L 4 charge −1. Each of η k carries a charge ω k 8 of Z 8 and then η 0 , η 4 can be real. The fields η 1 and η 3 are necessarily present and are connected by CP µτ as The rest of the fields, η 2 , η 0 , η 4 , transform trivially under CP µτ [12] Then the neutrino Yukawa couplings will be where, due to CP µτ , y N e , c ee and c µτ are real while y N τ = y * N µ , c τ τ = c * µµ and c eτ = c * eµ . The Dirac mass matrix will be diagonal as where m D = v y N e is real by symmetry and κ = |κ| can be made real and positive by rephasing L α . The heavy neutrino mass matrix will have the CP µτ symmetric form where e.g. A = c ee η 0 . We assume that CP µτ is preserved by η k , i.e., Light neutrino masses will be generated by the seesaw mechanism as M ν = −M T D M −1 R M D , whose inverse is closely related to M R as We get the texture-zero a = 0 or b = 0 if either η 0 or η 4 is absent and that is inherited from texture-zeros in M R in the same positions (A = 0 or B = 0). When solutions exist to accommodate the oscillation data, the matrix M −1 ν is completely fixed, except for experimental error. We show the possible solutions in Sec. III. And then, M R will depend only on two free parameters, m D , κ, as We will use m D or y N e interchangeably as one of the free parameters. Concerning mixing angles, it is guaranteed that any matrix in the form (1), which is symmetric by CP µτ , can be always diagonalized by a matrix of the form [2,3] where u i are all real and positive. Moreover, the Majorana type diagonalization (also known as Takagi factorization) will already lead to a real diagonal matrix and only discrete choices of signs -the CP parities-will appear instead of Majorana phases. In this way, the mass matrices for the light and heavy neutrinos can be diagonalized as where U (0) ν and U R are in the form (15), and the primed masses denote m i = ±m i and M i = ±M i , with m i and M i being the actual light and heavy masses. The complex conjugation in U (0) R appears because it is defined as the transformation matrix for N R whereas M R is defined in the basis N c R N c R . So Eq. (16) implies that the full diagonalizing matrices can be written as where K ν , K R are diagonal matrices of 1 or i depending on the signs on (16a) or (16b), respectively. Since a sign flip of both M ν and M R is not physical, we can distinguish four discrete cases of CP parities according to the sign of the diagonal entries of K 2 ν [8] as As we seek texture-zeros, some cancellation between m i will be necessary and the case (+ + +) will not appear in our solutions. The generic possibilities for K 2 R as well as the detailed mass spectrum and mixing pattern will be discussed in Sec. IV. Opposite parities in K 2 R will also give rise to cancellations in the CP asymmetries of heavy neutrinos suppressing the resonant enhancement.
We limit ourselves here to discussing briefly the limit κ = 1, which is straightforward. Considering (14) and since we can identify With this equation fixing the ordering for (M 1 , M 2 , M 3 ) in (16b), we have the direct relation This means that the spectrum for the heavy neutrinos is completely fixed in terms of the light masses and the CP parities for the heavy neutrinos are opposite to those of the light neutrinos. Therefore, K 2 R = −K 2 ν and As κ deviates from unity, U R will deviate from U (0) ν depending only on the parameter κ. The same will happen for the mass ratios between two heavy masses. Only the absolute scale for M i will be controlled by m D (or y N e ).

III. LIGHT NEUTRINOS
The inverse of the light neutrino mass matrix in the flavor basis is CP µτ symmetric and was given in (1) with a or b possibly vanishing. Different texture-zeros are not phenomenologically possible because it would lead to vanishing θ 13 (or also θ 12 ) [12]. Since M ν itself is CP µτ symmetric, the usual predictions of maximal θ 23 = 45 • and δ = ±90 • follow as θ 13 = 0 [2,3].
Without texture-zeros, the five parameters in (1) -a, b, |c|, |d|, arg(d 2 c * )-should describe the remaining five observables not fixed by symmetry: θ 12 , θ 13 , m 1 , m 2 , m 3 . Among these five observables, only four combinations are currently experimentally determined and we cannot predict the only unknown quantity: the lightest neutrino mass (equivalently, the absolute neutrino mass scale). With the additional one-zero texture, the number of free parameters is reduced by one and all the observables can be fixed, including the lightest neutrino mass. We show the possible solutions in table I when we allow for the experimental uncertainties for observables not fixed by symmetry, in accordance to the global-fit in Ref. [23] 3 . The procedure to find these solutions are explained below. A relatively wide range for m 1 appears for case II because it is a merger of two discrete solutions that would appear if there were no experimental error.  We can see that case V has too large masses and it is excluded by the Planck power spectrum limit (95% C.L.) [24], We are left with two cases for the normal ordering (NO) and two cases for the inverted ordering (IO). All these cases are also compatible with the latest KamLAND-Zen upper limit for the neutrinoless double beta decay parameter at 90%C.L. [25], The variation in the latter, comes from the uncertainty in the various evaluations of the nuclear matrix elements. In the future, experiments such as KamLAND-Zen 800 will probe the IO region that includes our case IV and possibly our case III. The solutions in table I are obtained with the expressions for a, b in terms of physical parameters, which we show below. To derive them, we first choose the parametrization for the PMNS matrix, without Majorana phases, as where, e.g., c 13 = cos θ 13 , and we are choosing the Dirac CP phase to be e iδ = −i following the current hints from global fits [23]; the opposite Dirac CP phase can be used by taking the complex conjugate of (25). Note that the standard parametrization corresponds to diag(1, 1, −1)U (0) ν diag (1, 1, +i). The parametrization in (25) obeys the CP µτ symmetric form (15) but with the additional rephasing freedom from the left fixed by the choice Re(U [12]. This phase convention implies a certain phase relation between c and d in (1). With that phase convention in mind, (16a) is still guaranteed [3].
If we invert the relation (19) by using (25), we can write the parameters a, b, c, d in terms of the neutrino inverse masses and mixing angles: Choosing e iδ = +i instead, would correspond to taking d → d * and c → c * . Note that the phases of c, d in (26) follow a specific phase relation characterized by the compatibility between necessary for the consistency of (19). The rephasing freedom in (6) changes the phases of c and d accordingly. Other relations between the parameters in (1) and the physical parameters can be extracted from Ref. [12] by replacing . For example, a rephasing invariant measure of CP violation is given by which is nonzero in all physical cases. We would obtain the same result with opposite sign if we had e iδ = i. Finally, with the expressions for a and b in hand, we can seek solutions for a = 0 or b = 0 depending on the CP parities in (18).
As a further prediction of our scenario, various correlations between measured and unmeasured observables are expected due to the reduced number of parameters. We show in Fig. 1, for cases I, II and III in table I, the correlation between sin 2 θ 12 and the yet to be measured effective parameter which controls the neutrinoless double beta decay (0νββ) rates induced by light neutrino exchange. For case IV, such a correlation is weak and we show in Fig. 2 the correlation between m ββ and |∆m 2 It is clear that a better measurement of sin 2 θ 12 (|∆m 2 3− |) will lead to a sharper prediction of m ββ for cases I, II and III (case IV). In special, for case II, it is predicted that sin 2 θ 12 0.325 and for case IV, m ββ is within reach of the future experiments such as KamLAND-Zen.

IV. HEAVY NEUTRINOS
Here we show the spectrum and the mixing pattern of heavy neutrinos. We denote the states with definite masses by N i , i = 1, 2, 3. All parameters of the mass matrix for light neutrinos were determined in Sec. III and the discrete possibilities were listed in table I. Then all the information on the heavy neutrino mass matrix follows from (14). There are only two free parameters: m D (or y N e ) and κ. The first will set the overall scale for the heavy neutrino masses M i , i = 1, 2, 3, while κ will determine the mass ratios and mixing pattern. Note that we will not follow the usual convention where (N 1 , N 2 , N 3 ) are ordered from lighter to heavier states and then it is useful to denote the lightest heavy neutrino as N 0 and its mass as M 0 .
We can continue the analysis of the case of κ = 1, which we started in Sec. II. In this case, Eq. (14) implies that the heavy neutrino mass matrix is proportional to the inverse of the light neutrino mass matrix and the diagonalizing matrix is completely fixed by the PMNS matrix; cf. (20). The values of the heavy masses are completely determined by (21), except for an overall scale. From the solar mass splitting we always have M 2 < M 1 and the ratio is fixed by For the NO solutions of table I, at most a mild hierarchy of M 1 /M 2 ∼ 3.6 is expected. In contrast, for IO, m 1 is not the lightest mass and it is more useful to rewrite For both cases III and IV, M 1 is only about 1.5% larger than M 2 and the pair N 1 -N 2 is nearly degenerate. The ordering for M 3 , on the other hand, depends on whether the ordering follows the NO or IO: From these relations, a hierarchy of at most M 1 /M 3 ∼ 20 or M 3 /M 1 ∼ 50 is possible for NO or IO, respectively. The least hierarchical case, M 1 /M 3 ∼ 2, is possible for case II. We see that the lightest mass is M 0 = M 3 for NO and M 0 = M 2 for IO. The mixing matrix U R is also fixed by (20) for κ = 1. The first row of U R should have values The CP parities of the heavy neutrinos are also fixed by the relation (21): they are opposite to the CP parities of light neutrinos, i.e., When κ deviates away from unity, the mass spectrum will cease to obey Eqs. (30) or (32) and U R will no longer obey (20). Nevertheless, we can still establish that −K 2 R and K 2 ν should have the same signature, i.e., they are the same except for possible permutations. The proof is shown in appendix B. The result is that a clever choice of ordering for M i allows us to maintain (34). A possibility is to order the heavy neutrinos in such a way that (21) is valid when we continually take the limit 4 to κ = 1. In the same limit, U R should approach U (0) ν . With this ordering convention, 4 In practice (16b) isolates the eigenvalue M i that have the unique CP parity [−(K 2 R ) ii < 0] because the massless case never occurs. The remaining M i of the same sign never cross and they can be tracked unambiguously; see discussion around (37).
Obviously, only the second set is allowed for texture-zero solutions in Table I. We show how the heavy neutrino spectrum depends on κ in Fig. 3 for NO (cases I and II) and in Fig. 4 for IO (cases III and IV) by plotting the possible values for the heavy masses M i relative to the lightest mass M 0 | κ=1 at κ = 1. We clearly see that the mass spectrum obeys (30) [or (31)] and (32) for κ = 1. To make the plots, we diagonalize M R in (14) explicitly, keeping the convention in (35), and vary the observables not fixed by symmetry within their 3-σ values reported in Ref. [23] by random sampling. Then the minimal and maximal values are extracted to draw the borders. 5 We also indicate the CP parities for each N i and we see that the convention in (35) is enough to separate M 1 from M 2 for both cases II and III. For case IV, it seems that M 1 and M 2 cross near κ = 1 but one can check by varying only κ that they never cross. The minimal value of |M i − M 0 | for this case is checked to be 1.2% of M 0 = M 2 . An alternative way to gain analytic information of the heavy masses from the light neutrino masses are shown in appendix C.
FIG. 3: Mass spectrum for NO solutions (cases I and II) relative to the lightest mass M0 for κ = 1, for N1 (orange), N2 (green) and N3 (blue), as a function of κ. We use the 3-σ ranges in Ref. [23] for the observables not fixed by symmetry. M i indicate the heavy neutrino masses with their CP parity.
FIG. 4: Mass spectrum for IO solutions (cases III and IV) relative to the lightest mass M0 for κ = 1, for N1 (orange), N2 (green) and N3 (blue), as a function of κ. We use the 3-σ ranges in Ref. [23] for the observables not fixed by symmetry. M i indicate the heavy neutrino masses with their CP parity.
We can prove generically that when their CP parities are included no crossing of eigenvalues M i occurs when κ is continuously changed. The proof utilizes the rephasing invariant in (28) adapted to M R when parametrized as (11): The diagonalizing matrix U (0) R * is parametrized as (25) after appropriate rephasing of the second and third rows, and the respective angles are replaced as θ ij → Θ ij with upper case C ij , S ij denoting e.g. C ij = cos Θ ij . 6 Then the relation (14) allows us to conclude that i.e., it never vanishes due to (28). Hence M i never cross.
We can now turn to the mixing matrix U R . To show how the mixing matrix U R deviates from iU ν for κ = 1, we need a parametrization for U R . We use the decomposition in (17) and the parametrization in (15). Two among the three entries u i = |U R1i | in the first row are enough to recover the entire matrix U (0) R [2]. The procedure is reviewed in appendix D. Their behavior can be seen in Fig. 5 for the NO cases and in Fig. 6 for the IO cases. The limit for κ = 1 is clearly in accordance with (33) except for case IV where the rapid variation for κ near unity makes it hard to ascertain the value of |U Re1 | and |U Re2 | at the exact point. We have checked that they agree with (33).

V. LEPTOGENESIS
The SM cannot explain the present baryon asymmetry of the Universe expressed in the present abundance [24]: where n B is the baryon number density and s is the entropy density. When the SM is extended through some form of seesaw mechanism to account for naturally small neutrino masses, leptogenesis arises as a natural mechanism to explain the baryon asymmetry [7]. In the simplest type I seesaw mechanism, a lepton number asymmetry is generated when the lightest heavy Majorana neutrino typically decays more to antileptons than leptons due to CP violating 6 We use the convention that U R diagonalizes M * R and not M R . The equality (20) implies that Θ ij = θ ij for κ = 1.

FIG. 5: Modulus of
UR1i for our NO solutions (cases I and II) for i = 1, 2, 3 (orange, green, blue) as a function of κ. We use the 3-σ ranges in Ref. [23] for the observables not fixed by symmetry.
FIG. 6: Modulus of UR1i for our IO solutions (cases III and IV) for i = 1, 2, 3 (orange, green, blue) as a function of κ. We use the 3-σ ranges in Ref. [23] for the observables not fixed by symmetry.
Yukawa couplings. This lepton number asymmetry is then converted, within the SM, to a baryon asymmetry by spharelon processes that violate B + L but conserve B − L [26]. The CP asymmetries in the decays of N i depend on the Yukawa couplings λ iα that control the strength of the Yukwawa interactionsN iφ † L α , in the basis where M R is diagonal. In our model, we simply have where N R = U R N R in our convention and y N e can be used insted of m D . Due to the highly constrained nature of our setting, only two free parameters govern the heavy neutrino sector. We follow the ordering convention from the κ = 1 limit and recall that the lightest heavy neutrino is denoted by N 0 and its mass by M 0 . The two free parameters, y N e and κ, cannot vary completely without limit as perturbativity of Yukawa couplings requires roughly that This requirement typically furnishes lower and upper values for κ. For example, if the lightest heavy neutrino mass is M 0 = 10 12 GeV, we will be restricted to 10 −2 κ 10 2 . For lower M 0 , the allowed range increases proportionally to M −1/2 0 . See Eq. (56) in the following.
In the context of CP µτ symmetric models, it is known for some time that leptogenesis induced by singlet heavy neutrinos cannot proceed in the one-flavor regime where T ∼ M 0 10 12 GeV [3]; see also Ref. [8]. The reason is that CP µτ restricts the flavored CP asymmetries (0) α in the decay N 0 → L α + φ to obey [8] (0) Hence, the total CP asymmetry vanishes, and a net lepton number asymmetry cannot be generated. Only in the flavored regime [27,28] where the τ flavor can be distinguished by fast Yukawa interactions, i.e., when 10 9 GeV T ∼ M 0 10 12 GeV, leptogenesis can be successful in generating enough lepton number asymmetry [8]. See Ref. [29] for a recent analysis of the temperature regimes where the various SM interactions enter in equilibrium. Below 10 9 GeV, where all lepton flavors can be distinguished, Ref. [8] concluded within analytical approximations that leptogenesis cannot proceed because the washout in the µ and τ flavors are equal, so that the asymmetries (41) in these flavors are summed to zero. So our case is a particular case of purely flavored leptogenesis [30] with the distinction that the vanishing of (0) is protected by CP µτ and not by B − L. It is also a particular case, enforced by symmetry, of a case where the baryon asymmetry is generated only by the low energy Dirac CP phase and no CP violation is present in the heavy neutrino sector [31]. The equality of the washout effects for µ and τ flavors follows because, in the approximation where off-shell ∆L = 2 scatterings and off-diagonal correlations through the A-matrix are neglected, these washout effects are controlled by the three washout parametersm where v = 174GeV in the SM and the subscript 0 refers to N 0 . With CP µτ symmetry, and the strength of washout is the same in the latter flavors [8]. In our model, this fact can be directly checked for (39). Current neutrino parameters implies that typicallym 0 = αm 0α m * ≈ 1.07 meV and N 0 reaches the equilibrium density rather quickly and a strong washout of lepton flavors takes place depending onm 0α m * . The mass m * ≡ 16π 2 v 2 u 3M pl g * π 5 quantifies the expansion rate of the Universe. So we focus on the intermediate regime where 10 9 T ∼ M 0 10 12 GeV and neglect the possible asymmetries generated by the decay of heavier N i . We comment on possible effects in the end. In this regime, only the τ Yukawa interactions are in equilibrium and then only the τ flavor and its orthogonal combination are resolved by interactions. In this case, the final baryon asymmetry can be approximated by [32] where µ ,m 02 =m 0e +m 0µ , and the efficiency factor is valid for the strong washout regime but allows weak or mild washout in some flavor [28]. The factors 417/589 and 390/589 correspond to the diagonal entries of the A matrix and quantifies the effects of flavor in the washout processes when changing from the asymmetry in lepton doublets to asymmetries in ∆ α = B/3 − L α [28]; see also [32]. We ignore the small effects of off-diagonal elements of the A matrix and consider the third family Yukawas in equilibrium as well as h c . We can see that the properties (41) of CP µτ leads to a partial cancellation of the baryon asymmetry in (45) but it is nonzero because the τ flavor and its orthogonal combination are washed out differently. The quantity Y eq N0 is the equilibrium thermal density of N 0 per total entropy density and is given by Y eq N0 = 135ζ(3) 4π 4 g * ≈ 3.9 × 10 −3 , where the last numerical value is for the SM degrees of freedom below the N 0 mass (g * = 106.75). The factor 28/79 corresponds to the reduction of the asymmetry in ∆ α to B − L in the SM due to spharelons when they go out of equilibrium before EWPT.
In the CP µτ symmetric case, we can rewrite (45) in the form where we denote One can note that the sign of the final baryon abundance is determined by the sign of − (0) τ because the combination η τ − η 2 > 0, as the washout function (46) is a decreasing function in the strong washout regime wherem 0 > m * .
The necessary CP asymmetry in the τ flavor, in the generic type I seesaw case, can be written as The part proportional to f (x), the vertex function, corresponds to the one-loop vertex contribution while the rest corresponds to the self-energy contribution for N R . We are assuming that N j masses are hierarchical, i.e., |M j −M 0 | Γ 0 for N j different from the lightest one and the N 0 decay width is It is easy to see that for κ = 1, the flavored CP asymmetry (49) is vanishing as (λλ † ) ij ∝ δ ij due to our simple form (39). Therefore, at least a small departure from κ = 1 is necessary to obtain a nonzero abundance. In fact, the expression in (49) can be simplified to The full expression is shown in appendix E. We can now analyze how the different quantities depend on our free parameters κ and y N e . It is clear from (14) and (39) that M R and λ iα scale as y 2 N e and y N e , respectively. Then mass ratios M i /M 0 andm 0α in (43) are independent of y N e and only depend on κ. On the other hand, the CP asymmetry in (49) scales as y 2 N e and that is also the scaling behaviour of the baryon abundance in (47). Therefore, the only dependence of Y ∆B on y N e can be factorized as y 2 N e while the remaining expression only depends on κ.
It is much more convenient, however, to consider the lightest heavy mass M 0 as the free parameter instead of y N e , for each κ. We can trade y N e for M 0 as follows. First, we factor the dependence of the lightest eigenvalue of M R on κ with fixed y N e by defining The masses M i are calculated from the eigenvalues of (14) with fixed y N e , say y N e = 1. Generically, f 0 (κ) is a monotonically increasing (hence one-to-one) function with f 0 (1) = 1 but not smooth when there is a crossing of M i (differently for M i which never cross). This function can be seen in the blue band of Fig. 3 for NO where M 0 = M 3 for all κ. The band is due to the variation within 3-σ of the low energy observables not fixed by symmetry. 7 For IO, M 0 = M 2 or M 0 = M 3 depending on κ for case III and always M 0 = M 2 for case IV. The function f 0 is shown in the low-lying green-orange (green) band of Fig. 4 for case III (IV). The transition from M 0 = M 2 to M 0 = M 3 for case III leads to discontinuities in λ 0α due to reordering of U Rα0 ; see Fig. 5. These in turn, lead to jumps inm 0α for this case.
As a second step, we define a reference value for M 0 : where m max is the heaviest light neutrino mass: m 3 for NO and m 2 for IO. The dependence of M 0 on κ and y N e can be made explicit as The inverse relation then gives y N e as a function of M 0 for each κ: Hence, y N e is completely determined for each M 0 (scaling as √ M 0 ) and κ. For example, the perturbativity requirement in (40) can be easily extracted. The relation (56) and the function (53) accomplish the purpose of expressing all the relevant quantities in the baryon asymmetry (47) solely in terms of κ and M 0 . Moreover, the dependence on M 0 is only multiplicative as Using (53) we can write, for example, the explicit dependence on κ of whereλ is the Yukawa matrix with y N e factored out, i.e., We have checked that typicallym 02 ,m τ > 20 meV and strong washout in all flavors take place. Only for case IV, m 0τ ∼ 0.5-0.6 meV for κ > 1 and the asymmetry in the τ flavor is washed out only mildly. The N 0 decay width can be also rewritten as This relation allows us to check that we will be typically away from the resonant regime because for M 0 = 10 12 GeV and our four solutions in table I. Lower values of M 0 will give proportionally lower ratios. We can now show in Fig. 7 the baryon asymmetry Y ∆B we expect for our four solutions, considering M 0 = 10 12 GeV and δ = −90 • for the low-energy Dirac CP phase. Results for lower values of M 0 can be reinterpreted by rescaling linearly as in (57) down to M 0 ≈ 10 9 GeV where the flavor regime with τ resolved ceases to be valid. We also show −Y ∆B (dashed style and darker colors) which corresponds to the baryon asymmetry for the disfavored case δ = 90 • , because flipping the sign of δ flips the signs of both (0) τ and Y ∆B . For the current preferred value of δ = −90 • , only cases I, III and IV can give the right asymmetry in certain parameter regions, some of them very narrow. The value δ = +90 • is disfavored in more than 3σ in current global fits [23] and case II is then the least favored. The possible parameter regions in the κ-M 0 plane that can lead to successful leptogenesis are shown in table II where only the rectangular borders enclosing the real regions are listed. These regions can be read off from Fig. 7. For example, for case I, only the region around κ ≈ 8 and M 0 ≈ 10 12 GeV survives because for a lower value of M 0 , the red region will be scaled down proportionally and a sufficient asymmetry cannot be generated. In all cases for δ = −90 • , successful leptogenesis requires that M 0 be restricted to the narrow band of the intermediate region:   Few comments are in order. Firstly, and most surprisingly, Fig. 7 shows no divergent resonant peak for case III where the two lightest heavy masses approach the degenerate limit near κ = 1, albeit our use of the CP asymmmetry (49) which do not include any regulator [33]. The reason is that in our model the CP asymmetry (0) τ do not diverge for heavy neutrino masses of opposite CP parity even in the degenerate limit because the divergence in the vertex correction is cancelled by the self-energy contribution. See appendix E for the explicit expression. This feature explains the lack of divergenes in Fig. 7 and also applies to the CP asymmetry of the heavier N i . For case IV, there is indeed a peak near κ = 1 but there is no divergence because M 1 − M 2 never really vanish. The minimal value of |M i − M 0 |/M 0 = 1.2% implies that we do not reach the resonant regime and no regulator is needed since the width is much smaller; cf. (61).
Secondly, we note that our results for successful leptogenesis listed in table II should not be interpreted as precise values but rather as rough estimates. The approximate formula (46) we used for the final efficiency factor has an estimated uncertainty of the order of 30% [28]. Some neglected effects such as thermal corrections and spectator processes may also lead to small corrections; see e.g. [32]. We also assumed that at a temperature of 10 12 GeV the τ Yukawa interaction is already fast enough that the τ flavor can be distinguished from the rest but, in reality, there is a transition region where some correlation among flavors may survive until 10 11 GeV [29]. In this transition region, correlations that are off-diagonal in flavor may be important.
Another important aspect in our case is the possible effect of the heavier N i in the generation and washout of additional lepton asymmetry for temperatures T > M 0 . As can be seen in Figs. 3 and 4, there are regions for the solutions for case I and case III in which the hierarchical approximation is justified. But in other regions, the masses M i are not hierarchical and the effects of heavier N i may not be negligible; see Refs. [32,34,35] and references therein. In fact, for all cases, there are large ranges for κ where the ratio between the second lightest and the lightest mass is less than 10. The mass difference may even vanish (almost vanish) for case III (IV) as discussed above. However, as the window for successful N 0 leptogenesis is already narrowly restricted between 10 11 GeV and 10 12 GeV, the decay of the heavier N i will not generate a lepton asymmetry if the latter is generated above 10 12 GeV where there is no flavor effect and the total asymmetry vanishes due to (41), still valid in this case. Some lepton asymmetry may be generated below 10 12 GeV, but we have checked that the CP asymmetry generated by the decay of the second lightest N SL into τ flavor is at most of the same order of (0) τ and the total washout parameter is large,m SL m * , although the parameter for τ flavor could be smaller than unity. So, these effects are at most of the same order and a detailed account is beyond the scope of this paper.
With the previous caveats in mind, it is worth discussing the case where the real effenciency factor is actually 10% smaller than our approximation in (46). In this case, the region for case I and and the second region for case III disappear completely, leaving only two regions of IO as viable solutions for δ = −90 • . Moreover, for case IV, only a narrow region near κ = 1 is allowed and in this region the heavy neutrino parameters are approximately determined. For example, the heavy masses M i are approximately proportional to m −1 i ; cf. (21). In contrast, no new regions appear if the efficiency factor were 10% larger.

VI. CONCLUSIONS
We have shown a highly predictive model of leptons where the light neutrino sector is completely determined up to discrete solutions and the heavy neutrino sector responsible for the seesaw is controlled by only two free parameters.
The model implements the µτ -reflection symmetry in the neutrino sector and its predictions of maximal atmospheric angle, maximal Dirac CP phase, and trivial Majorana phases follow. The model allows both the maximal values ±90 • for the Dirac CP phase but the negative value is currently preferred from global fits. The predictivity is increased by additionally enforcing an abelian Z 8 symmetry, combined nontrivially with the µτ -reflection symmetry, that leads to one texture zero in the (ee) or (µτ ) entry of the heavy neutrino mass matrix and hence transmitted to the inverse of the light neutrino mass matrix. No free parameters are left in the low energy theory after the neutrino observables are accommodated and only four solutions for the lightest neutrino mass are possible depending on three possible CP parity combinations. The possible values are shown in table I. There are two solutions for normal ordering and two solutions for inverted ordering. In all cases, except one, the lightest neutrino mass lies in the few meV range. Only in one NO solution, the lightest mass can vary up to 30 meV. The effective parameter that controls neutrinoless double decay through light neutrino exchange is completely fixed as well. One of the solutions for IO is within reach of the KamLAND-Zen experiment in the 800 phase which will probe the IO region [36]. Due to the reduced number of parameters, correlations between the neutrinoless doublet beta decay parameter m ββ and other oscillation observables arise.
In parallel, the two free parameters of the heavy neutrino sector completely control the mass spectrum and the mixing relative to the charged leptons. One parameter sets the overall mass scale and the other controls the mass hierarchy and mixing angles. The heavy neutrino sector is then further constrained from the requirement of successful leptogenesis. Only small regions in the space of the two free parameters are allowed. These regions can be seen in table II. For the preferred value of δ = −90 • , only three out of the four solutions, one NO and two IO, allow the production of enough baryon asymmetry. In all cases, the lightest heavy neutrino mass needs to lie roughly in the small window of 10 11 to 10 12 GeV where flavor effects are crucial. Since the window is narrow, the maximal amount of generated baryon asymmetry is sensitive to the efficiency factor that quantifies the washout effects and even a 10% reduction would eliminate the NO solution and only two small regions with IO would remain. Moreover, in our model, a resonant enhancement of the CP asymmetry is not possible if the degenerate heavy neutrinos have opposite CP parities and all our CP asymmetries are finite even without the inclusion of a regulator.
In summary, a highly predictive model of leptons is presented where all parameters of the theory, except two, are completely fixed. These two parameters in turn controls the heavy neutrino sector and are further constrained to small regions from successful leptogenesis.
where we can decompose w k as w k = |w k |e iγ k . (D2) The modulus and relative phases of the second and third rows can be obtained from orthogonality as where γ ji ≡ γ j − γ i . The quadrant ambiguity of γ ij can be resolved by either one of the unitary triangles |u 1 w 1 | + |u 2 w 2 |e iγ21 + |u 3 w 3 |e iγ31 = 0 , The individual γ i are most easily calculated in the phase convention where D in (11) is real and recall that U (0) R diagonalizes M * R in our convention. In this case, the eigenvector equation leads to , Du i sin γ i = −|w i |(B sin 2γ i + Im C) .

Appendix E: Simplified CP asymmetry
The full expression of the simplified CP asymmetry (52) in the τ flavor is We describe briefly in the following how to obtain it. We stress that there is no resonant enhancement if K 2 R00 K 2 Rjj = −1, i.e., if N 0 and N j have opposite CP parity, because the combination −g(x) + 1/(1 − x) approaches −1/2 + ln(4) ≈ 0.88 in the limit x → 1.