Combining texture zeros with a remnant CP symmetry in the minimal type-I seesaw

In the framework of the two right-handed neutrino seesaw model, we consider maximally-restrictive texture-zero patterns for the lepton Yukawa coupling and mass matrices, together with the existence of a remnant CP symmetry. Under these premisses, we find that several textures are compatible with the most recent data coming from neutrino oscillation and neutrinoless double beta decay experiments. It is shown that the number of allowed texture zeros depends on the type of heavy right-handed neutrino mass spectrum. Namely, at most one (two)-zero patterns are allowed in the Dirac Yukawa coupling matrix for nondegenerate (degenerate) heavy Majorana neutrinos. The predictions for the low-energy Dirac and Majorana CP-violating phases, and for the effective mass parameter relevant in neutrinoless double-beta decay experiments, are presented and discussed. We also comment on the impact of future experimental improvements in scrutinising texture-zero patterns with a remnant CP symmetry, within the minimal version of the seesaw mechanism considered here.


Introduction
The observation of neutrino oscillations, which require the existence of neutrino masses and mixing, provides an irrefutable evidence for physics beyond the Standard Model (SM). Experimentally, three lepton mixing angles and two neutrino mass-squared differences have been precisely measured [1][2][3]. Also, hints for a nonzero Dirac CP-violating phase have been found [4,5], being the best-fit value obtained from global analysis of neutrino oscillation data close to 3π/2 [1][2][3]. On the other hand, the Majorana character of neutrinos (which would imply the existence of Majorana phases) is still to be scrutinised, with the most promising efforts coming from searches for neutrinoless double-beta (0νββ) decays [6][7][8]. Furthermore, cosmological data and beta-decay experiments, indicate that neutrino masses are at least six orders of magnitude smaller than charged-lepton masses, suggesting that neutrinos may be particles of a different nature.
In the context of SM extensions, the seesaw mechanism [9] offers a natural explanation for neutrino mass suppression. In the most straightforward seesaw realisation, heavy righthanded (RH) neutrinos are added to the SM field content (type-I seesaw), leading to an effective Majorana neutrino mass term. While these models describe qualitatively well the neutrino oscillation parameters, predicting them quantitatively becomes very difficult without further constraining the underlying model. In general, adding seesaw mediators introduces a large number of parameters at high energies, when compared to the number of low-energy observables potentially measurable by experiments. To maximize predictability, without compromising the data requirement of three nonzero mixing angles and at least two nonzero neutrino masses, two RH neutrinos must be added to the SM particle content. This is referred to as the two RH neutrino seesaw model (2RHNSM). Nevertheless, even in such minimal type-I seesaw framework, the restrictions on the high-energy Lagrangian are not sufficient to unambiguously determine the low-energy neutrino parameters.
It is widely established that flavour and charge-parity (CP) symmetries may strongly constrain fermion masses and mixing . In spite of being a demanding task [18], imposing CP invariance in a consistent flavour framework is appealing since it may potentially lead to nontrivial constraints on both Dirac and Majorana CP-violating phases, and, ultimately, on 0νββ decay. When generalised flavour and/or CP symmetries of the theory are broken, a set of remnant CP symmetries may still be preserved by the charged-lepton and/or neutrino sectors of the Lagrangian [21,25,26], for energies below the breaking scale. The impact of remnant CP invariance, both at high and low energies, has already been studied in the context of the 2RHNSM [31]. In particular, it has been shown that the 3 × 2 orthogonal matrix O that establishes a connection between high and low energy parameters in the 2RHNSM (in the so-called Casas-Ibarra parametrisation [32]) depends on a single real parameter in the presence of a remnant CP symmetry.
When texture zeros (which may be motivated by flavour symmetries [33][34][35][36][37][38][39][40]) are considered in the Yukawa coupling and mass matrices of the 2RHNSM, the low-energy CPviolating phase space is substantially reduced [41][42][43][44][45][46][47]. In this context, the presence of a texture zero in the Dirac neutrino Yukawa coupling matrix fixes the single complex parameter in O, in terms of low-energy observables [46]. Combining texture zeros with a remnant CP symmetry will, in principle, lead to more stringent relations among the neutrino parameters. Furthermore, predictions for the CP-violating Dirac and Majorana phases may be obtained, which could be tested in future neutrino experiments [48]. Thus, a thorough study of texture zeros in the presence of remnant CP symmetries is called for.
In this work, we study the 2RHNSM with one remnant CP symmetry in the neutrino sector when the maximum number of texture zeros is imposed on the lepton Yukawa and mass matrices. We analyse all possible texture zero patterns in the light of current neutrino oscillation data. The layout of the paper is as follows. In section 2, we briefly review the scenario of a single remnant CP symmetry in the neutrino sector of the general type-I seesaw model. Using the Casas-Ibarra parametrisation, we analyse the restrictions stemming from imposing that symmetry and apply them to the 2RHNSM. In section 3, we identify the maximally restricted texture-zero cases and study their compatibility with current neutrino data. The predictions for the low-energy CP-violating phases and the effective mass parameter relevant in 0νββ decay experiments are given in section 4. Finally, in section 5, we summarise our results and present the concluding remarks.

Remnant CP symmetry in the type-I seesaw model
Adding N RH neutrino fields ν R to the SM particle content, leads to the leptonic Lagrangian L = L + L ν + L CC , with Here, L ≡ (ν L , e L ) T is a left-handed (LH) lepton doublet, Φ ≡ (φ + , φ 0 ) T is the SM Higgs doublet, e R are the RH charged-lepton fields, and W ± µ are the charged gauge boson fields. The 3 × 3 (3 × N ) general complex matrix Y (Y ν ) is the charged-lepton (Dirac neutrino) Yukawa coupling matrix, while M R stands for the complex symmetric N ×N RH neutrino mass matrix. The effective neutrino mass matrix M ν , generated after electroweak symmetry breaking, is given by the type-I seesaw formula [9], where v ≡ φ 0 174 GeV. Being symmetric, M ν can be diagonalised by the unitary matrix U ν , such that where m i are the real and positive light neutrino masses. Rotating the LH neutrino and charged-lepton fields to their physical basis by ν L → U ν ν L and e L → U e L , a flavour misalignment arises in the charged-current interactions L CC . Lepton mixing is then encoded in the unitary matrix U, where θ ij (i < j = 1, 2, 3) are the lepton mixing angles (with s ij ≡ sin θ ij , c ij ≡ cos θ ij ), δ is the Dirac CP-violating phase and  [2,3]). Since the sign of ∆m 2 31 is not yet determined, the results are shown for the two possible neutrino mass orderings: Inverted ordering (IO): Despite providing a very elegant and consistent way of explaining neutrino masses, the type-I seesaw model lacks in predictability. This can be easily concluded by counting the parameters in the neutrino sector. The Dirac Yukawa coupling matrix Y ν is described by 6N real parameters, namely, 3N moduli and 3N phases, from which three can be removed after redefining the LH charged-lepton fields. Adding the N RH neutrino masses, the highenergy neutrino sector is thus described by 7N −3 physical parameters. On the other hand, at low energies, the effective mass matrix M ν is defined in terms of the three light neutrino masses, three mixing angles and three phases, for a total of nine physical parameters. Thus, for N ≥ 2 (required to accommodate neutrino data), the number of low-energy parameters is insufficient to uniquely determine the high-energy Lagrangian of the theory.
In the charged-lepton mass basis, a convenient way of parametrising Y ν is through the Casas-Ibarra parametrisation [32], where d M ≡ diag(M 1 , · · · , M N ) and M i are the real and positive heavy neutrino masses M i , obtained after diagonalising M R with a unitary matrix U R , such that Obviously, in the heavy neutrino mass basis M R = d M , U R = 1 and, from eq. (2.11), M . Also, O is an orthogonal 3 × N complex matrix parametrised by two real parameters for N = 2, and by 6(N − 2) real parameters for N ≥ 3 [50]. Since any matrix O obeying OO T = 1 leads to the same low-energy M ν , determining O implies considering further assumptions.
Imposing CP invariance of the theory provides an attractive way of increasing predictability. At high energies, the lepton sector of a given model may be invariant under a generalised CP symmetry [13] which, after spontaneous symmetry breaking, may result into remnant CP symmetries of the charged-lepton and neutrino sectors. In this work, we assume that a single remnant CP symmetry of the form is preserved by the Lagrangian of eq. (2.1). Here, ν c L,R ≡ Cν T L,R , being C the chargeconjugation matrix. The transformation matrices X ν and X R are 3 × 3 and N × N unitary complex matrices, respectively. In case the Lagrangian of eq. (2.1) is invariant under the above symmetries, Y ν and M R must satisfy the conditions which, together with eq. (2.4), imply From eqs. (2.5) and (2.17), one may also see that U ν obeys where in the last equality we have taken into account the fact that the light neutrinos are nondegenerate (the ± signs in the above equation are independent). In such case, besides being unitary, X ν is also symmetric [15]. If one of the neutrinos is massless, X ν has the more general form where the presence of the phase factor e iφ , instead of ±1, is due to m 1 = 0 (m 3 = 0) for NO (IO). For a given U ν , eq. (2.18) fixes the CP transformation matrix X ν as X ν = U * ν X ν U † ν which, in the charged-lepton mass basis takes the form X ν = U * X ν U † . For RH neutrinos, using eqs. (2.12) and (2.16), we have where X R is automatically unitary, due to the unitarity of U R and X R . In contrast to what is observed for light neutrinos, degeneracies in the heavy Majorana neutrino mass spectrum must be taken into account. For n ≤ N degenerate RH neutrinos, i.e. M 1 = M 2 = ... = M n , the solution of the above equation is where O n is a n × n general real orthogonal matrix, and D N −n is a (N − n) × (N − n) diagonal matrix with entries ±1, i.e. D N −n = diag(±1, ±1, · · · , ±1). In general, X R is symmetric only if n = 0 (nondegenerate heavy neutrinos In what follows, we will consider both degenerate and nondegenerate heavy neutrino mass spectra. For nondegenerate heavy neutrinos, it can be easily shown that the remnant CP symmetry enforces the matrix elements of O to obey the relation O ij = ±O * ij , i.e., O ij are either real or purely imaginary. Therefore, in this case, the number of free parameters in O is decreased by half, being now equal to one if N = 2, and 3(N − 2) if N ≥ 3. On the other hand, when all heavy neutrinos are degenerate, i.e. n = N in eq. (2.21), one has from eq. (2.22) This means that the matrix O in the Casas-Ibarra parametrisation must be such that O N is a real matrix, since X R is unitary. Obviously, if X ν = ±diag(1, 1, 1), O is constrained to be real, and O N = ±1 N . In this case X R = ±1 N and, as expected, the CP transformation in the RH neutrino sector is trivial.

The two RH neutrino case
We now consider the 2RHNSM, i.e., the minimal type-I seesaw scenario compatible with experimental data. In this case, two heavy Majorana neutrinos, with masses M 1 and M 2 , are responsible for the neutrino mass generation, being one of the light neutrinos massless (m 1 = 0 for NO, or m 3 = 0 for IO). Moreover, one of the Majorana phases in eq. (2.8) can be removed by redefinition of the neutrino fields, leaving a single physical CP-violating Majorana phase α. In our notation, this corresponds to replacing P in eq. (2.7) by Hence, the number of low-energy neutrino parameters in the 2RHNSM is reduced to seven: three mixing angles, two nonzero light neutrino masses, a Dirac phase and a single Majorana phase. On the other hand, at high energies, Y ν is a 3×2 matrix defined by nine independent parameters, while M R is a 2 × 2 matrix with two parameters (in the RH neutrino mass basis). Furthermore, the matrix O in eq. (2.11) is an orthogonal 3 × 2 complex matrix parametrised by a complex angleẑ in the following way [42] NO with ξ = ±1 resulting from a discrete indeterminacy in O. As previously seen, the presence of CP symmetries may decrease the number of free parameters in a generic seesaw model. In the specific case of the 2RHNSM, X ν and X R are given by eqs. (2.19) and (2.21), respectively, where N = 2 must be taken. For nondegenerate heavy neutrinos (M 1 = M 2 ), imposing one remnant CP symmetry decreases the number of free parameters in Y ν by one (see discussion at the end of the previous subsection). Therefore, the total number of high-energy parameters is reduced to ten, which is still large when compared to the seven low-energy observables. According to eq. (2.21), for M 1 = M 2 , one has X R = diag(±1, ±1). Since the transformations X ν → − X ν and X R → − X R leave eq. (2.22) invariant, we will only consider the cases X R = diag(1, ±1) in our analysis. All the allowed configurations for O, obeying eq. (2.22) with different forms of X ν and X R , are presented in table 2 (see also ref. [31]). As expected, for M 1 = M 2 , the elements O ij are either real or purely imaginary, and only a single real parameter z is required to define O. For the degenerate case Since O 2 must be real, the above relation restricts O to be of the type O I given in table 2 for (a, b) = ±(1, 1) in X ν (for notation, see the table caption or eq. (2.19)). On the other hand, if (a, b) = ±(1, −1), O 2 is automatically real for any matrix O. In this case, no constraints are imposed by the CP symmetry and the number of parameters in Y ν remains unchanged. In summary, except for the case discussed above with M 1 = M 2 and (a, b) = ±(1, −1), the remnant CP symmetry forces the elements of O to be either real or purely imaginary. Thus, O is parametrised by a single real parameter z (instead of two in the general 2RHNSM), and Y ν is described by eight independent parameters (instead of nine in the general 2RHNSM). In order to increase the degree of predictability of the 2RHNSM com-bined with a CP symmetry, in the following we will consider maximally restricted texture zeros in Y , Y ν and M R .

Texture zeros in the 2RHNSM with a remnant CP symmetry
In a previous work [46], we studied maximally restricted texture zero patterns for Y , Y ν and M R in the 2RHNSM. We concluded that the maximal number of texture zeros that can accommodate neutrino and charged-lepton data is six in Y , two in Y ν and a single zero in M R . In this scenario, only an IO light-neutrino mass spectrum turns out to be compatible with data. For each of the viable patterns, it was shown that the lowenergy Dirac and Majorana phases can be expressed in terms of the well-measured mixing angles and neutrino mass-squared differences, being the results independent of the mass ratio r N = M 2 /M 1 . We now intend to analyse the compatibility of those textures when a remnant CP symmetry is imposed.
In the basis where Y is diagonal, the following textures for M R will be considered: where '×' denotes a generic nonvanishing entry and '·' indicates the symmetric nature of the matrix. Notice that R 4 automatically leads to M 1 = M 2 , being the only nondiagonal pattern allowing for a mass degeneracy of the two heavy neutrinos. Therefore, only R 1 and R 4 may reproduce a degenerate RH neutrino mass spectrum. Let us first discuss the possibility of having two texture zeros in Y ν together with the remnant CP symmetry. The possible Y ν patterns in this case are [46] T 1 : Taking into account the results obtained in ref. [46], the following conclusions hold in the present work: • For M R of the type R 4 , it was shown that none of the above T textures are compatible with neutrino data. This is true regardless of the presence of the CP symmetry.
, O is not constrained by the CP symmetry and the results from ref. [46] hold. Namely, only T 1 , T 2 , T 4 and T 5 are viable for the IO case, being δ close to the current best-fit value δ 3π/2 and α 1.9π (0.08π) for T 1 and T 4 (T 2 and T 5 ).
• For R 1 with M 1 = M 2 and (a, b) = ±(1, 1), and R 1,2,3 with M 1 = M 2 , the CP symmetry constrains O to be parametrised by a single real parameter. For those cases, it will be shown later that all the above T patterns are excluded by data. In view of the above, and for those cases in which two texture zeros are excluded, we will now turn our attention to the one-texture zero Y ν patterns Combining M R of the type R 4 (M 1 = M 2 ) with the above textures using eq. (2.4), we obtain the M ν patterns (A, D and F in the notation of ref. [46]) shown in the second column of table 3. Notice that, in all cases, M ν features one texture zero in a diagonal entry, which lead to testable relations among the low-energy parameters (see eq. (2.5)). As concluded in ref. [46] and shown in the third and fourth columns of table 3, texture D (F) is compatible with neutrino data at 3σ for an IO neutrino mass spectrum 1 with δ and α varying in the ranges [−0.3, 0.3]π ([0.8, 1.2]π) and [0.8, 1.2]π, respectively. These results remain valid in the present framework as long as the remnant CP symmetry does not introduce further constraints on the model parameters, which is the case for (a, b) = ±(1, −1) (unconstrained O). In conclusion, only textures Y 2 , Y 5 , Y 3 and Y 6 are viable when M R is of the R 4 type and (a, b) = ±(1, −1). The remaining texture combinations with M 1 = M 2 (M R of type R 1,4 ) and (a, b) = ±(1, 1) will be discussed later.
Let us now study the case M 2 = M 1 , for which only patterns R 1 , R 2 and R 3 need to be considered. As we shall see later, the most restrictive case of two texture zeros in Y ν and the existence of a remnant CP symmetry in the 2RHNSM is not compatible with 1 A given texture combination is considered compatible with data at 3σ when all predicted parameters lie in the 3σ range given in table 1 and at least one is out of the 1σ interval. current neutrino data. Therefore, we analyse the patterns of Y ν with one texture zero. For all combinations (R 1,2,3 , Y 1−6 ), no zeros (or any other constraint) arise in M ν , so that the presence of a single texture zero in Y ν does not constrain the low-energy parameter space. Nevertheless, imposing a texture zero in Y ν leads to the determination ofẑ in eq. (2.25) in terms of the low-energy neutrino parameters [46] (and M 1,2 for a non-diagonal M R ), fixing the matrix O in eq. (2.11). Thus, the number of independent parameters in Y ν is reduced from nine to seven. For instance, in the simplest case of diagonal Y and M R , the constraint ( Y ν ) 11 = 0 together with eq. (2.11) and O given by eq. (2.25), leads to NO: where cẑ ≡ cosẑ and sẑ ≡ sinẑ. The above relations determineẑ as Although a single texture zero in Y ν does not lead to low-energy constraints in the general 2RHNSM, this is not the case when a remnant CP symmetry is considered. In this framework (see section 2.1), and according to eq. (2.22), the matrix O is parametrised by a real parameter z, reducing the number of parameters in Y ν from seven to six. Hence, we are left with eight high-energy parameters to be compared with seven low-energy observables. A summary of the parameter counting in Y ν taking into account the restrictions imposed by the existence of texture zeros and/or remnant CP symmetries in the 2RHNSM is presented in table 4 for M 1 = M 2 . The number of constraints in Y ν and the relevant/predicted physical parameters are also presented for each case. 2 Depending on the constraints imposed, some of these parameters can be written in terms of the remaining ones and low-energy predictions are then obtained.
We now turn to the compatibility analysis of the patterns given in eq. (3.3) when M R exhibits the forms R 1,2,3 . Depending on the column in which the zero in Y ν is placed, the following conditions are obtained where U R , defined in eq. (2.12), depends solely on the ratio r N = M 2 /M 1 , and Y ν corresponds to Y ν defined in the heavy neutrino mass basis. Using the parametrisation (  one has which lead to the relations among the low-energy neutrino parameters and r N given in 1+rν sin(2θ 12 ) Table 5. Relations among low-energy parameters obtained in the 2RHNSM with one remnant CP symmetry, for Y ν and M R of the forms Y 1−6 and R 1−3 , respectively. Here, r N = 1 and O = O I . All expressions are invariant under α → 2π − α and δ → 2π − δ, which is reflected in figures 1 and 2. Notice that, since interchanging the second and third lines of Y ν corresponds to the same operation in U (see eq. (2.11)), the results for Y 3 (Y 6 ) are obtained from those of Y 2 (Y 5 ) performing the transformation θ 23 → θ 23 + π/2. This can be seen by comparing the second (sixth) and third (seventh) rows of the table. These properties are general and, therefore, are also verified in table 6. Due to the complexity of the relations for O I , only the zeroth order in s 13 is shown for the results of R 2,3 with the patterns Y 2,3,5,6 . The cases marked with '' are excluded due to the condition tan z = i (z is not real).
the condition | tanh z| < 1 must also be ensured. This implies further constraints among the relevant parameters, 3 which may or may not be compatible with those of table 6. For illustration, let us consider the simplest combination (R 1 ,Y 1,4 ), for which the condition     1), the matrix O is constrained to be real and of the type O I (see discussion after eq. (2.26)), the results in table 5 are valid for all combinations (R 1 , Y 1−6 ). If, instead, M R is of the type R 4 , the texture-zero conditions (3.8) and (3.9) imply tan z = ±i, which is incompatible with the requirement of O being real. Therefore, all combinations (R 4 ,Y 1−6 ) are excluded for (a, b) = ±(1, 1).
To conclude this section, let us briefly comment on the compatibility of two texturezero patterns for Y ν in the present framework. For the NO case, none of such textures is compatible with neutrino data, even in the absence of the CP symmetry [46]. This remains true for IO when r N = 1 and the CP symmetry is imposed, as can be seen by combining the results in tables 5 and 6 for two different patterns of Y ν with zeros in distinct columns [46]. For instance, when M R is diagonal and Y ν is of the form T 1 , the low-energy constraints are those coming from Y 1 and Y 5 simultaneously. As indicated in the second and fourth rows of table 5, this corresponds to implying (α, δ) = (2nπ, nπ), with n ∈ Z. These solutions are not in the (α, δ) regions allowed for T 1 [46]. Proceeding in the same way for the remaining patterns with M 1 = M 2 , and M 1 = M 2 with (a, b) = ±(1, 1), it can be shown that two texture zeros in Y ν are not compatible with neutrino oscillation data in the presence of a remnant CP symmetry.

Parameter space analysis
In this section, we will investigate the compatibility with data of the 2RHNSM with maximally-restricted texture zeros in the presence of a remnant CP symmetry. Our analysis is focused on the predictions for leptonic CP violation and 0νββ decay. Having discussed all two-texture zero cases in the previous section, we will now analyse the viability of the one texture zero M R and Y ν patterns, given in eqs. (3.1) and (3.3), respectively, for nondegenerate RH neutrinos. 4

Leptonic CP violation
Taking into account the data of table 5, and of table 6 combined with the condition | tanh z| < 1, we now present the allowed regions in the (α, δ) plane for each Y ν and M R textures, considering the 3σ and 1σ experimental ranges for the neutrino mixing angles and mass-squared differences (see table 1 NO (upper plots) and IO (lower plots), 5 considering ξ = ±1. Since for O I the results depend on the mass ratio r N , we consider the representative values r N = 3, 10, 100 (blue, purple and pink regions, respectively). Notice that, for r N 1, the regions are very similar to those obtained for diagonal M R , where there is no dependence on r N (cf . table 5). For Y 4 , this is easily understood taking the limit r N 1 in the expressions of table 5, while for Y 5,6 the complete expressions would be needed. Moreover, from comparison of the results in figures 1 and 2 for O II and O III (in red and green), it can be seen that the same allowed regions appear for R 1 and R 2 , which is explained by the results given in table 6. Up to now, we have not taken into account the experimental constraints on the CPviolating phase δ, due to its poor statistical significance at present. Varying this phase within the experimental range given in table 1, we can study the allowed regions in the space of the only two free parameters in the model, namely, α and r N . The results presented in figure 3 show that, in most cases, r N and α are strongly correlated and nontrivial bounds on r N can be set. Moreover, for the most part, the allowed range for r N may be very narrow, depending the value of α.

Neutrinoless double beta decay
When the only relevant contributions to 0νββ decays are due to the exchange of light neutrinos, 6 the corresponding rates are sensitive to the effective mass for NO and IO. These expressions, together with the experimental ranges for the neutrino parameters given in table 1, lead to the allowed regions in the (α, m ββ ) plane presented in figure 4. To illustrate the present experimental sensitivity of 0νββ experiments, the upperlimit intervals for m ββ coming from the CUORE [52] and KamLAND-Zen [53] experiments are also shown. Taking the 1σ and 3σ ranges for current neutrino oscillation data, the following m ββ ranges are obtained: partially covers the IO ranges. On the other hand, the NO scenario will only be probed by new-generation projects (discussions on present results and prospects for neutrinoless double beta decay experiments can be found in refs. [6][7][8]).
Given the above results, we will now analyse the impact on m ββ of the previously discussed texture-zero patterns and remnant CP symmetry. In figures 5 and 6, we show the predictions for m ββ when the constraints given in tables 5 and 6 (and the condition | tanh z| < 1) are imposed on α and the neutrino data of table 1 is used. The blue, red and green points correspond to the cases in which O is of the type O I , O II and O III , respectively. Furthermore, darker (lighter) colours stand for the predictions obtained taking the 1σ (3σ) intervals for all parameters given in table 1. The results for R 1 are presented in figure 5 for the Y ν textures Y 1−6 , while for R 2 they are given in figure 6 for Y 4−6 . In the latter figure, ξ = ±1 is considered, and only the case O = O I is shown, since O II and O III lead to the same predictions as those with R 1 (see figure 5). Moreover, r N varies within the compatibility regions shown in figure 3. For comparison, we include in both figures the delimiting lines of the NO and IO regions of figure 4, valid for the general 2RHNSM. Again, the results for R 3 are related to those of R 2 , as commented in footnote 5. When compared to the general 2RHNSM, the results of figures 5 and 6 show that, imposing texture zeros in Y ν and M R , together with a remnant CP symmetry, can have a profound impact on  Degenerate RH neutrinos (r N = 1)  We conclude this section by remarking that our results are valid in the basis in which the charged-lepton Yukawa matrix Y is diagonal. When Y is non-diagonal (but still contains six texture zeros), the analysis can be performed following the same procedure of ref. [46], i.e., by applying column and/or row permutations on the textures considered here for the diagonal case.

Conclusions
In the framework of the 2RHNSM, we have studied all maximally restricted texture-zero patterns for lepton Yukawa and mass matrices in the presence of the remnant CP symmetry defined in eqs. (2.13) and (2.14). Predictions for the CP-violating phases δ and α, as well as for the effective neutrino mass parameter m ββ (relevant for neutrinoless double beta decay), were obtained and confronted with the present experimental data. In tables 7 and 8, we present a texture compatibility summary for degenerate and nondegenerate RH neutrinos, respectively. Our results show that: • For degenerate RH neutrinos (M R of the form R 1 or R 4 ) the remnant CP symmetry does not have any impact on the allowed parameter space when that symmetry is such that (a, b) = ±(1, −1) (see eqs. (2.13), (2.19) and the discussion around eq. (2.26)). Therefore, in these cases, all constraints stem from the texture zeros imposed on Y ν and both one-or two-texture-zero Y ν patterns are allowed for IO (see table 7). As explained in section 3, the predictions for δ, α and m ββ are those presented in ref. [46]. On the other hand, for (a, b) = ±(1, 1), the matrix O is constrained to be real, implying additional restrictions on the low-energy parameters. The compatible textures in this case are also indicated in table 7.
• For nondegenerate RH neutrinos (M R of the form R 1 , R 2 or R 3 ), the matrix O is parametrised by a single real parameter when the remnant CP symmetry is imposed (see table 2). This implies an additional constraint on the low-energy neutrino parameters, besides those coming from the existence of texture zeros in Y ν . In this case, all two-texture-zero Y ν patterns given in (3.2) are excluded by data, for both NO and IO. With one texture zero in Y ν , the compatible patterns are indicated in table 8.
• In general, the restrictions on the low-energy parameters for the viable one-texturezero Y ν patterns indicated in table 8 are stronger when M R is diagonal (see figures [1][2][3][4][5][6]. This is due to the fact that, with M R of the type R 2 or R 3 , α and m ββ are sensitive to the mass ratio r N . In turn, this parameter is also constrained by data, as shown in figure 3. Notice that, in all cases, and depending on the value of α, a nontrivial lower bound on r N can be set. For instance, r N 30 in the case (R 2 , Y 5 , ξ = −1).
Our findings show the importance of measuring low-energy neutrino parameters with precision for scrutinising texture-zero patterns in the 2RHNSM, when a remnant CP symmetry is imposed. In this sense, future improvements on the measurement of the phase δ by neutrino oscillation experiments like T2K [4] and NOνA [5], and on neutrinoless double beta decay rates [6][7][8] will further constrain the scenarios analysed in this paper.  Table 8. Compatibility summary for the 2RHNSM in presence of the remnant CP symmetry defined in eqs. (2.13) and (2.14). All cases with Y ν of the type Y 1−6 or T 1−6 , M R of the forms R 1−3 , and nondegenerate heavy Majorana neutrinos are indicated. The compatibility for the R 2,3 cases is verified for at least one value of r N .