Predictive Dirac and Majorana Neutrino Mass Textures from $SU(6)$ Grand Unified Theories

We present simple and predictive realizations of neutrino masses in theories based on the $SU(6)$ grand unifying group. At the level of the lowest-dimension operators, this class of models predicts a skew-symmetric flavor structure for the Dirac mass term of the neutrinos. In the case that neutrinos are Dirac particles, the lowest-order prediction of this construction is then one massless neutrino and two degenerate massive neutrinos. Higher-dimensional operators suppressed by the Planck scale perturb this spectrum, allowing a good fit to the observed neutrino mass matrix. A firm prediction of this construction is an inverted neutrino mass spectrum with the lightest neutrino hierarchically lighter than the other two, so that the sum of neutrino masses lies close to the lower bound for an inverted hierarchy. In the alternate case that neutrinos are Majorana particles, the mass spectrum can be either normal or inverted. However, the lightest neutrino is once again hierarchically lighter than the other two, so that the sum of neutrino masses is predicted to lie close to the corresponding lower bound for the normal or inverted hierarchy. Near future cosmological measurements will be able to test the predictions of this scenario for the sum of neutrino masses. In the case of Majorana neutrinos that exhibit an inverted hierarchy, future neutrinoless double beta experiments can provide a complementary probe.


I. INTRODUCTION
Multiple neutrino oscillation experiments over the past two decades have conclusively established that neutrinos have non-vanishing masses [1], thereby providing concrete evidence of new physics beyond the Standard Model (SM). However, although these experiments have measured the neutrino mass splittings and mixing angles, the actual values of the neutrino masses still remain unknown. In particular, it is not known whether the neutrino mass spectrum exhibits a normal or inverted hierarchy. Several medium and long-baseline neutrino oscillation experiments have been proposed to settle this issue [2]. At present, the important question of whether neutrinos are Dirac or Majorana fermions also remains unanswered. Future neutrinoless double beta decay (0νββ) experiments may be able to resolve this question [3].
Grand unification [4][5][6] is one of the most attractive proposals for physics beyond the SM.
In these theories, the strong, weak and electromagnetic interactions of the SM are unified into a larger grand unifying group. The fermions of the SM are embedded into representations of this bigger group, with the result that quarks and leptons are also unified into the same multiplets. These representations often contain additional SM singlets, which can naturally serve the role of right-handed neutrinos in the generation of neutrino masses. The fact that the SM quarks and leptons are now embedded together in the same multiplets often leads to relations between the masses of the different SM fermions [7]. If these multiplets also contain right-handed neutrinos, these theories can impose restrictions on the form of the neutrino mass matrix, leading to predictions for the neutrino masses. Familiar examples of unified theories that can relate the masses of the neutrinos to those of the charged fermion include the Pati-Salam [4] and SO(10) [8,9] gauge groups.
In this paper we explore a class of models based on the SU (6) grand unified theory (GUT) [10,11] that lead to sharp predictions for the neutrino mass spectrum. In these theories, the right-handed neutrino emerges from the same multiplet as the lepton doublet of the SM. A natural consequence of this construction is that, at the level of the lowestdimension terms, the Dirac mass term for the neutrinos is skew-symmetric in flavor space, so that the determinant of the Dirac mass matrix vanishes. If neutrinos are Dirac particles that obtain their masses from this term, then, in the absence of corrections to this form from terms of higher dimension, the neutrino mass spectrum consists of two degenerate species and a massless one. Once higher-dimensional terms suppressed by the Planck scale M Pl are included, this class of models can easily reproduce the observed spectrum of neutrino masses and mixings. A firm prediction of this construction is that the spectrum of neutrino masses is inverted, with the lightest neutrino hierarchically lighter than the other two. Then the sum of neutrino masses is predicted to lie close to the lower bound of 0.10 eV set by the observed mass splittings in the case of an inverted hierarchy. Future precision cosmological experiments, such as LSST [12], Euclid [13], DESI [14], the Simons Observatory [15], and CMB-S4 [16], that have the required sensitivity to the sum of neutrino masses will be able to test this striking prediction. The final phase of Project-8 [17], with an expected sensitivity of 0.04 eV to the absolute electron neutrino mass, will also be able to test this scenario.
Similarly, future large-scale long-baseline neutrino oscillation experiments, such as Hyper-K [18] and DUNE [19], will be able to test the prediction regarding the inverted nature of the mass spectrum.
It is well-established that there is a lower bound on the light neutrino contribution to the 0νββ process in the case of Majorana neutrinos that exhibit an inverted masshierarchy [20,21]. In particular, it has been pointed out that if long-baseline neutrino experiments determine that the neutrino mass hierarchy is inverted, while no signal is observed in 0νββ down to the effective Majorana neutrino mass m ee 0.03 eV, then this would constitute compelling evidence that neutrinos are Dirac rather than Majorana fermions [22].
The model we present here is an example of a GUT framework that can naturally accommodate such a scenario.
If, in addition to the skew-symmetric Dirac mass term, there is also a large Majorana mass term for the right-handed neutrinos, the neutrinos will be Majorana particles. In this scenario, the skew-symmetric nature of the Dirac mass term implies that the lightest neutrino is massless, up to small corrections from higher-dimensional operators. In contrast to the case of Dirac neutrinos discussed above, the spectrum of neutrino masses can now exhibit either a normal or inverted hierarchy. However, the lightest neutrino is still predicted to be hierarchically lighter than the other two, so that for both normal and inverted hierarchies the sum of neutrino masses is predicted to lie close to the corresponding lower bound dictated by the observed mass splittings, i.e. 0.06 eV for the normal case and 0.10 eV for the inverted. This is a prediction that can be tested by future cosmological observations once long-baseline experiments have determined whether the spectrum is normal or inverted. In addition, these predictions for the sum of neutrino masses translate into upper and lower bounds on the 0νββ rate for each of the normal and inverted cases, with important implications for future 0νββ experiments. In our analysis, we explore both the Dirac and Majorana possibilities in detail and obtain realistic fits to the observed masses and mixings.
To understand the origin of the prediction that the Dirac mass term for the neutrinos is skew-symmetric, we first consider the minimal grand unifying symmetry, namely SU (5) [5].
In this class of theories the SU (5) grand unifying symmetry is broken at the unification scale, M GUT ∼ 10 16 GeV, down to the SM gauge groups. In simple models based on SU (5), all the SM fermions in a single generation arise from the5 and 10 representations. The5 is the anti-fundamental representation while the 10 is the tensor representation with two antisymmetric indices. The Higgs field of the SM is contained in the fundamental representation, the 5. The up-type quark masses arise from Yukawa couplings of the schematic form κλµνρ 5 Hκ 10 λµ 10 νρ , where 5 H contains the SM Higgs, κλµνρ is the 5-dimensional antisymmetric Levi-Civita tensor, and the Greek letters represent SU (5)  One possible solution to this problem, first explored in Refs. [29,30], is that the third-generation up-type quarks emerge in part from the 20 of SU (6), which is the tensor representation with three antisymmetric indices. This decomposes as 20 → 10+10 under SU (5).
This allows the third-generation up-type quarks to obtain their masses from a renormalizable term of the form κλµνρσ 6 Hκ 15 λµ 20 νρσ . Nonrenormalizable operators suffice to generate masses for the up-type quarks of the lighter two generations.
The problem of the top quark mass in SU (6) GUTs admits an alternative solution if electroweak symmetry is broken by two light Higgs doublets rather than one, so that the low-energy theory is a two-Higgs-doublet model. In this framework, one of Higgs doublets, which gives mass to the up-type quarks, is assumed to arise from the 15 of SU (6). This allows all the up-type quark masses to be generated from renormalizable terms of the form The right-handed neutrinos can naturally acquire large Majorana masses of order M 2 GUT /M Pl ∼ 10 14 GeV from nonrenormalizable Planck-suppressed interactions with the Higgs fields that break the GUT symmetry. This naturally leads to Majorana masses for the neutrinos of the right size through the seesaw mechanism [31][32][33][34]. Alternatively, as a consequence of additional discrete symmetries, a Majorana mass term for the right-handed neutrinos may not be allowed, while the coefficient of the Dirac mass term is suppressed. In such a scenario we obtain Dirac neutrino masses. In this paper we will consider both the Dirac and Majorana cases.
This paper is organized as follows. In Section II, we outline the framework that underlies this class of models and show how the pattern of neutrino masses emerges in the Dirac and Majorana cases. In Section III, we present a realistic model in which the neutrino masses are Dirac, and perform a detailed numerical fit to the neutrino masses and mixings using a recent global analysis of the 3-neutrino oscillation data. We show that this framework predicts an inverted spectrum of neutrino masses with one mass eigenstate hierarchically lighter than the others. In Section IV, we present a realistic model in which the neutrino masses are Majorana, and again perform a detailed numerical fit to the neutrino oscillation data. We show that in this scenario one neutrino is again hierarchically lighter than the others, but the spectrum of neutrino masses can now be either normal or inverted. We also explore the implications of this scenario for future 0νββ experiments and future cosmological observations. Our conclusions are presented in Section V.

II. THE FRAMEWORK
Our model is based on the SU (6) GUT symmetry with the fermions of each family arising from a6 representation, denoted by χ, and a rank-two antisymmetric representation 15, denoted by ψ. For now we omit the generation indices. Note that anomaly cancellation for the SU (6) group requires that there be two6 chiral fermion representations for each 15 fermion. We denote the additional6 of each family byχ. After the breaking of SU (6) to SU (5), the fields inχ that carry charges under the SM gauge groups acquire large masses at the GUT scale by marrying the non-SM fermions in the 15. Therefore, these fields do not play a role in generating the masses of the light fermions. However, the SM-singlet field inχ, which has no counterpart in the 15, may remain light. We employ the familiar convention in which all fermions are taken to be left-handed, and the SM fermions are labelled as (Q, u c , d c , L, e c ), with Q T = (u, d) and L T = (ν, ).
The SU (6) symmetry is broken near the GUT scale down to SU (5), which contains the usual embedding of SM fermions in a5 and a 10 of SU (5). Without loss of generality we take the SU (5) indices to be (2, 3, 4, 5, 6), so that the index 1 lies outside SU (5). Color indices run over (4,5,6).
We now consider the assignment of fermions under representations of SU (6). Under the fermion multiplet χ that transforms as a 6, we have where L is the SM lepton doublet, L T = (ν, ). Note that the Dirac partner ν c of the SM neutrino is embedded in the same multiplet as the left-handed leptons. The fermions inχ also transform as6: The fermion content of ψ, which transforms as a 15-dimensional representation of SU (6), is given by The breaking of SU (6) down to SU (5) at the GUT scale is realized by a Higgs fieldĤ which transforms as a 6 under SU (6) and acquires a large vacuum expectation value (VEV) along the SM-singlet direction. A Higgs fieldΣ, which transforms as an adjoint under SU (6), further breaks SU (5) down to the SM gauge group. The breaking of electroweak symmetry is realized through two Higgs doublets H and ∆ that arise from different SU (6) representations. The field H, which gives masses to the down-type quarks and charged leptons, emerges from a 6 while ∆, which gives masses to the up-type quarks, arises from a 15. The Higgs fieldsĤ, H and ∆ are assumed to have the following VEVs: The VEV ofΣ takes the pattern The field content is summarized in Table I. Here N F denotes the number of flavors.
We now discuss the generation of fermion masses. The additional fermionsL,D c inχ andL c ,D in ψ acquire masses at the GUT scale through interactions withĤ of the form where we have suppressed the SU (6) and Lorentz indices and shown only the flavor indices.
Consequently, these fields do not play any role in the generation of the masses of the SM fermions. These interactions do not give mass to the SM-singlet field N c inχ. However, even if N c is light, the fact that it is a SM singlet means that in the absence of other interactions its couplings to the SM fields at low energies are very small.
The SM fermions acquire masses from their Yukawa couplings to the Higgs fields H and ∆ after electroweak symmetry breaking. The SU (6)-invariant Yukawa couplings take the The down-quark and charged-lepton masses arise from the y d term in the Lagrangian after the Higgs field H acquires an electroweak-scale VEV. Similarly the up-quark masses arise from the y u term in the Lagrangian after ∆ acquires a VEV. In general, the masses of the SM fermions also receive contributions from higher-dimensional operators suppressed by the Planck scale (M Pl ) that involveΣ, such as The VEV ofΣ breaks the SU (5) symmetry that relates quarks and leptons [cf. Eq. (5)].
Therefore these higher-dimensional operators violate the GUT symmetries that relate the masses of the down-type quarks to those of the leptons of the same generation.
A Dirac mass term for the neutrinos may be obtained from interactions of the form As explained earlier, the fact that ∆ is an antisymmetric tensor under SU (6) implies that y ν,ij is skew-symmetric in flavor space. Consequently, the resulting Dirac mass matrix for the neutrinos has vanishing determinant. We expect corrections to the Dirac mass term from Planck-suppressed higher-dimensional operators, such as In general, this contribution will be suppressed by a factor M GUT /M Pl ∼ 10 −2 relative to that from Eq. (9).
A large Majorana mass term for the right-handed neutrinos can be obtained from Plancksuppressed nonrenormalizable interactions of the form This leads to Majorana masses for the right-handed neutrinos of order M 2 GUT /M Pl , which is parametrically of order the seesaw scale ∼ 10 14 GeV. Then, from Eqs. (9) and (11), we obtain Majorana neutrino masses of the right size.
If neutrinos are to be Dirac particles, the mass term for the right-handed neutrinos shown in Eq. (11) must be absent. Furthermore, we require the coefficients of the Dirac mass terms to be extremely small, y ν,ij , κ ν,ij ∼ 10 −11 , to reproduce the observed values of the neutrino masses. In Section III, we shall show that the absence of the Majorana mass term for the right-handed neutrinos, Eq. (11), and the smallness of y ν,ij and κ ν,ij can be explained on the basis of discrete symmetries.  (7) and (8), are consistent with the Z 4 and Z 7 symmetries. The interaction in Eq. (6) that gives GUT-scale masses to the extra fermionsL,D c inχ andL c ,D in ψ is also allowed by the discrete symmetries. However, the renormalizable Dirac mass term for the neutrinos, Eq. (9), is now forbidden by the discrete Z 7 symmetry. Instead, the leading contribution to the neutrino masses arises from the dimension-5 term The field σ, which is a singlet under SU (6), is assumed to acquire a VEV, thereby spontaneously breaking the discrete Z 7 symmetry. For σ ∼ 10 7 GeV, we obtain Dirac neutrino masses in the right range. Since ∆ is in an antisymmetric representation of SU (6), these mass terms are antisymmetric in flavor space, i.e.
This leads to a highly predictive spectrum, with one zero eigenvalue, and the other two eigenvalues equal in magnitude and opposite in sign. This corresponds to an inverted mass hierarchy, in which the smaller ∆m 2 arises from the difference between the masses of the two heavier eigenstates. We can perform phase rotations on the right-handed neutrinos to ensure that the elements of this mass matrix are real, so that the phase in the PMNS matrix vanishes.
Clearly, the mass pattern above is ruled out experimentally. However, we need to include the effects of higher-dimensional terms, which will give corrections to the pattern above.
Since these corrections are expected to be small, we expect to retain the qualitative features of the spectrum above, in particular, an inverted ordering. An example of such a higherdimensional operator is the dimension-6 term This correction is parametrically smaller than the antisymmetric contribution in Eq. (12) by a factor M GUT /M Pl ∼ 10 −2 .
In order for the terms in Eq. (12) to give rise to the leading contribution to the neutrino masses, other possible mass terms involving the light neutrino fields ν and ν c must be suppressed. The discrete Z 4 symmetry forbids Majorana mass terms for ν and ν c . It also forbids Dirac mass terms between ν and N c . A Dirac mass term between ν c and N c can be generated as a Z 7 -breaking effect, but only at dimension-8: This is too small to have any observable effect. Therefore, without loss of generality, the neutrino mass matrix has the form of a real skew-symmetric matrix with a small complex symmetric component. We write the mass term in matrix form as, Here M 0 ν is skew-symmetric and takes the form while δm is an anarchic symmetric matrix whose entries are parametrically smaller than those in M 0 ν . We can choose m a , m b and m c in Eq. (18) to be real without loss of generality. However, in general the elements of δm are complex.
The PMNS matrix U is, as usual, defined to be the rotation matrix that relates the flavor eigenstates ν of the active neutrinos to the mass eigenstates ν i : Defining D ν = diag(m 1 , m 2 , m 3 ) as the diagonalized mass matrix with mass eigenvalues m i corresponding to the eigenstates ν i , we have Therefore the PMNS matrix is identified with the matrix that diagonalizes the matrix M † ν M ν . By a suitable choice of of m a , m b , m c , and the elements in δm, we can fit the observed neutrino mass splittings and mixing angles.
Before proceeding with a numerical scan, we first estimate the region of parameter space consistent with observations. Although there are a large number of free parameters, since only m a , m b and m c are expected to be large, this scenario is very predictive. We parametrize the elements of the skew-symmetric matrix M 0 ν as follows: m a = m cos θ cos φ , zeroth order in this perturbation, the eigenvalues for M † ν M ν are simply {m 2 , m 2 , 0}. This corresponds to a limiting case of an inverted mass hierarchy in which the smaller (solar) mass splitting vanishes. By convention, in an inverted hierarchy the mass eigenstates m 1 , m 2 , m 3 are labeled such that m 3 corresponds to the mass of the lightest state and the smaller splitting is between m 1 and m 2 , with m 2 > m 1 . In our case, these correspond to the masses of two degenerate eigenstates with mass m. Then the eigenstate with vanishing mass is identified as ν 3 . The mixing angle θ 12 mixes states in the degenerate subspace, and hence is arbitrary at this order. It will be fixed by the perturbation. The other two mixing angles are given by θ 13 = θ and θ 23 = φ. The Dirac CP phase δ CP can be rotated away at this order as well.
To summarize, for δm = 0, which corresponds to zeroth order in the perturbation, the model predictions for the solar and atmospheric mass-squared splittings, the mixing angles, and the Dirac CP phase are given by ∆m 2 sol ≡ ∆m 2 21 = 0 , ∆m 2 atm ≡ |∆m 2 32 | = m 2 , θ 13 = θ , θ 23 = φ , θ 12 = arbitrary , δ CP = 0 , where ∆m 2 ij ≡ m 2 i − m 2 j . Once we add the perturbation δm, the solar splitting and the mixing angle θ 12 are fixed. The perturbation δm can be parametrized as η m, wherem is an anarchic symmetric matrix with entries of order m. The lightest eigenstate acquires a mass of order ηm from the perturbation, and the solar splitting is now The atmospheric mass splitting ∆m 2 atm ≡ |m 2 3 − m 2 2 | continues to remain of the order of m 2 . The ratio of the solar and atmospheric splittings determines the parametric size of η, which in turn sets the mass of the lightest eigenstate. Putting in the numbers, we have  We see that a satisfactory fit to the data requires the parameter η to be of order m 3 /m 1 ∼ 10 −2 . Remarkably, this is in excellent agreement with the expected value of η from our We see that this flavor pattern results in a very predictive spectrum of neutrino masses and mixings. We obtain an inverted mass hierarchy, with one neutrino hierarchically lighter than the other two. This prediction can be conclusively tested in future long-baseline oscillation experiments such as Hyper-K [18] and DUNE [19]. Since the CP -violating phase δ CP in the PMNS matrix vanishes in the limit that δm is zero, it might have been expected to be small.
However, the results of our numerical scans in Section III B show that this need not be the case, and that fairly large values of δ CP can be obtained even for η 10 −2 .  (Fit1, Fit2, Fit3).
The gray, green, and pink-colored contours represent the NuFit 1σ, 2σ, and 3σ CL allowed regions respectively, while the red markers represent the NuFit best-fit values for an IH. The blue, black, and brown markers are respectively the predictions of the benchmark points corresponding to Fit 1, Fit 2, and Fit 3, as given in Table IV.

B. Fits to the Data
Our strategy for the scan is as follows. The neutrino mass matrix is parameterized in terms of a skew-symmetric matrix M 0 ν with a small symmetric correction δm, as discussed in Section III A. We fix the parameters {m a , m b , m c } of the skew-symmetric matrix M 0 ν in Eq. (18) such that the zeroth order predictions match the measured values of ∆m 2 atm , θ 13 and θ 23 as given by Eq. (22). In particular, we take m 2 ≡ ∆m 2 atm = 2.509 × 10 −3 eV 2 , θ ≡ θ 13 = 8.61 • , and φ ≡ θ 23 = 48.3 • corresponding to the central values from NuFit [35] for the inverted hierarchy case and employ Eq. (21) to determine m a , m b , and m c . Further, the size of the perturbation η is fixed by ∆m 2 sol /∆m 2 atm . We then scan over the anarchic matrix m and obtain numerical predictions for the entire PMNS matrix. We choose to parametrize the mass matrix in Eq. (17) in terms of m c and the ratios x 1 ≡ m a /m c , x 2 ≡ m b /m c and As can be seen from Eq. The predictions of these fits for the oscillation parameters are shown in Table IV, along with the 3σ allowed range from NuFit4.1 global analysis [35]. Also included are the predictions for the mass of the lightest neutrino. Note that in each of these fits the lightest neutrino mass is hierarchically lighter than the other two mass eigenstates by more than two orders of magnitude. The results for the fits presented in Table IV are also displayed in Interestingly, we find no significant restriction on the CP -violating phase δ CP in the PMNS matrix in this scenario. In particular, as seen from Fit 3, we can get a large CP phase in the PMNS matrix even if all the elements of δm are smaller by a factor of order 10 −2 than the observed atmospheric splitting. Larger δ CP values seem to be preferred by the recent T2K results [36], and in the future, a more precise determination of δ CP can only help us better constrain the parameter space of the model. Yukawa couplings that generate masses for the SM quarks and charged leptons, Eqs. (7) and (8), are also allowed. Turning our attention to the neutrino sector, the renormalizable Dirac mass term for the neutrinos, Eq. (9), and the nonrenormalizable Majorana mass term for the right-handed neutrinos, Eq. (11), are both consistent with the discrete symmetry. In the absence of other mass terms involving ν and ν c , these interactions lead to the desired pattern of Majorana neutrino masses. The singlet neutrinos N inχ obtain large Majorana masses of order the right-handed scale through the operator The discrete symmetry forbids a renormalizable Dirac mass term between the SM neutrinos ν and the singlet neutrinos N . Any allowed Dirac mass terms between ν c and N are highly Planck suppressed and much smaller than their Majorana masses. It follows that the effects of N on the neutrino masses are small and can be neglected. Then, the Dirac mass term in Eq. (9) and the Majorana mass term in Eq. (11) give the dominant contributions to the neutrino masses, leading to Majorana neutrino masses of parametrically the right size that exhibit the pattern discussed in Section II.

B. Fits to the data
In this subsection, we obtain fits to the neutrino masses and mixings for the case of In the limit that M D M c ν , we can write the following seesaw relation for the light neutrino masses, |y 11 |e iϑ y 12 y 13 y 12 y 22 y 23 where we choose to parametrize the mass matrix in terms of and M 0 ≡ m 2 3 /M 33 . The overall mass scale M 0 is required to be tiny, of order 10 −11 GeV, to obtain the observed values of neutrino masses. We perform a numerical scan of the input parameters, as shown in Eq. (28), to obtain predictions for the entire PMNS matrix. It is beyond the scope of this work to scan over the full parameter space; instead, we perform a constrained minimization in which the five neutrino observables (sin 2 θ 12 , sin 2 θ 13 , sin 2 θ 23 , ∆m 2 21 ,  and |∆m 2 3l | with l = 1 in the case of normal hierarchy and l = 2 for inverted) are restricted to lie within 2σ of their experimentally measured values. The parameter M 11 has been chosen to be complex in order to induce a CP violating phase in the PMNS matrix, but the other parameters have been taken to be real. We emphasize that the lightest neutrino is exactly massless due to the skew-symmetric nature of the Dirac mass matrix M D .
The input parameters shown in Table VI provide excellent fits to the oscillation data, as can be seen in Table VII. For each of the benchmark points the CP phase in the PMNS and inverted hierarchy (IH) compared to our benchmark models for the Majorana case (Fit1, Fit2, Fit3, Fit4). The gray, green, and pink-colored contours represent the NuFit 1σ, 2σ, and 3σ CL contours respectively in the NH case, whereas the solid, dashed, and dotted lines correspond to the 1σ, 2σ, and 3σ CL contours respectively for IH. The red and purple markers in each case correspond to the NuFit best-fit values for the IH and NH respectively, while the blue, black, brown, and gray markers are the predictions of the benchmark models corresponding to Fit 1, 2, 3, and 4 respectively, as given in Table VII. In the bottom right panel, |∆m 2 3l | refers to the atmospheric mass-squared splitting, with l = 1 (2) for NH (IH). matrix is large, showing that there is no restriction on its value. Fits 1 and 2 correspond to an inverted hierarchy, whereas Fits 3 and 4 represent a normal hierarchy. The benchmark points (Fit 1, Fit 2, Fit 3 and Fit 4) are also displayed in Fig. 2 as Fit1 (IH), Fit2 (IH), Fit3 (NH), and Fit4 (NH) as blue, black, brown, and gray markers respectively in various two-dimensional projections of the global-fit results [35].
In the standard framework with only light neutrinos contributing to 0νββ, the amplitude for the 0νββ rate is proportional to the ee−element of the neutrino mass matrix, given by m ee = |m 1 c 2 12 c 2 13 + e iα m 2 s 2 12 c 2 13 + e iβ m 3 s 2 13 | .
Here m 1 , m 2 , and m 3 are the masses of the three light neutrinos, while s 2 ij ≡ sin 2 θ ij , c 2 ij ≡ cos 2 θ ij (for ij = 12, 13, 23), and (α, β) are the two unknown Majorana phases. We can apply Eq. (29) to our framework to determine its implications for 0νββ. Since the determinant of M D vanishes owing to its skew-symmetric structure, the lightest neutrino is exactly massless. For a given mass ordering (normal or inverted), the masses of the heavier two neutrinos can then be determined from the observed mass splittings. The expression for the effective Majorana mass given in Eq. (29) then reduces to one of the following equations, depending on whether the hierarchy is normal or inverted: m IH ee = |∆m 2 32 | − ∆m 2 21 c 2 12 c 2 13 + |∆m 2 32 | s 2 12 c 2 13 e iα .
Note that only one Majorana phase (or one specific linear combination of phases) is relevant, due to the smallest mass eigenvalue being zero.
To illustrate the range of possibilities for 0νββ in this class of models, in Fig. 3 we plot the effective Majorana mass as a function of sin 2 θ 12 , ∆m 2 21 and the sum of light neutrino masses m i . We restrict to points that lie within 1σ and 3σ of the allowed oscillation parameter range. Each data point in Fig. 3 represents a valid fit that has been obtained by scanning over the input parameters shown in Eq. (28). For the purposes of this scan, we have taken all the elements of the M ν c matrix to be complex. Here the blue (red) points correspond to the case of normal (inverted) hierarchy. The Majorana phases, as well as the other observables in Eqs. (30) and (31), have been obtained as predictions of the points in the scan. First, the PMNS matrix is identified with the matrix diagonalizing M † ν M ν , where M ν is given in Eq. (28). Then, taking U T M ν U = D ν gives the diagonalized mass matrix with the appropriate Majorana phases.
We can use Eqs. (30) and (31) to obtain upper and lower limits on the rate of 0νββ in this class of models. In the case of a normal hierarchy, the two terms in Eq. (30) add constructively for 0 ≤ (β −α) ≤ π/2, while partial cancellation occurs for π/2 ≤ (β −α) ≤ π.
The most effective cancellation (addition) happens when β − α = π (0  Future ton-scale 0νββ experiments such as LEGEND [37] and nEXO [38] should be able to probe the entire parameter space of this class of models if the hierarchy is inverted. For illustration, we show in Fig. 3  well within the 1σ sensitivity of CMB-S4, and so these measurements offer an opportunity to test this scenario.

V. CONCLUSION
In summary, we have presented a framework for neutrino masses in SU (6) GUTs that predicts a specific texture for the form of the leading contribution to the Dirac mass term.