Neutrino mass ordering and \mu-\tau reflection symmetry breaking

If the neutrino mass spectrum turns out to be m^{}_3<m^{}_1<m^{}_2, one may choose to relabel it as m^{\prime}_1<m^{\prime}_2<m^{\prime}_3 such that all the masses of fundamental fermions with the same electrical charges are in order. In this case the columns of the 3\times 3 lepton flavor mixing matrix U should be reordered accordingly, and the resulting pattern U^\prime may involve one or two large mixing angles in the standard parametrization or its variations. Since the Majorana neutrino mass matrix keeps unchanged in such a mass relabeling, a possible \mu-\tau reflection symmetry is respected in this connection and its breaking effects are model-independently constrained at the 3\sigma level by using current experimental data.


Introduction
In a simple extension of the standard electroweak model which can generate finite but tiny masses for the three originally massless neutrinos, the phenomenon of lepton flavor mixing measures a nontrivial mismatch between the mass and flavor eigenstates of the charged leptons and neutrinos. Similar to the dynamics of quark flavor mixing, the conjecture that the lepton fields interact simultaneously with the scalar and gauge fields leads to both lepton flavor mixing and CP violation in the standard three-flavor scheme [1]. The 3 × 3 lepton and quark mixing matrices which manifest themselves in the weak charged-current interactions are referred to as the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix U [2] and the Cabibbo-Kobayashi-Maskawa (CKM) matrix V [3], respectively: with all the fermion fields being the mass eigenstates. By convention U and V are defined to be associated with W − and W + , respectively. In Eq. (1) the charged leptons and quarks with the same electric charges all have the "normal" 1 and strong mass hierarchies [4], But for the time being it remains unclear whether the three neutrinos also have a normal mass ordering m 1 < m 2 < m 3 or not. Now that m 1 < m 2 has been fixed from the solar neutrino oscillation data by the 0 ≤ θ 12 ≤ π/4 convention and with the help of matter effects [5], the only possible abnormal neutrino mass ordering is m 3 < m 1 < m 2 , which has been referred to as the "inverted" mass ordering. Needless to say, the neutrino mass ordering is one of the central concerns in today's neutrino physics. A combination of current atmospheric (Super-Kamiokande) and accelerator-based (T2K and NOνA) neutrino oscillation data preliminarily favors the normal neutrino mass ordering up to the 2σ level [6,7]. But one has to be very cautious and open-minded at this stage, because only the next-generation neutrino oscillation experiments can really pin down the true neutrino mass ordering [8].
If the masses of three neutrinos end up in an inverted ordering, which is quite different as compared with the mass spectra of their nine charged partners in the standard model (see Figure 1 for illustration), one will have to explain or understand what is behind this "anomaly" from a theoretical point of view. From a purely phenomenological point of view, we may simply choose to relabel the three neutrino mass eigenstates in Eq. (1),  1 Here "normal" means that the charged fermions in the first family are the lightest and those in the third family are the heaviest when their masses are renormalized to a common energy scale, as shown in Figure 1. where the allowed ranges of three neutrino masses are quoted from Ref. [6], and the typical values of three charged-lepton masses and six quark masses are quoted from Ref. [9]. As for ν 1 , ν 2 and ν 3 , the horizontal cyan lines and red dashed lines stand for the normal mass ordering (NMO) and inverted mass ordering (IMO) cases, respectively. Note that the lower limit of m 1 in the NMO case or that of m 3 in the IMO case can be extended to zero. and thus make the mass spectrum become "normal": m ′ 1 ≡ m 3 < m ′ 2 ≡ m 1 < m ′ 3 ≡ m 2 , as shown in Figure 2. The invariance of the weak charged-current interactions (i.e., L cc ) under such a transformation requires that the PMNS matrix U transform accordingly, implying a striking change of the pattern of lepton flavor mixing. Provided U ′ is parametrized in the same way as U, the corresponding parameters must take different values. We shall show that U ′ may involve one or two large flavor mixing angles in the standard parametrization or its variations. We shall also illustrate that such a mass relabeling does not change the texture of the Majorana neutrino mass matrix M ν by taking into account a possible µ-τ reflection symmetry, and model-independently constrain its breaking effects at the 3σ level with the help of current neutrino oscillation data.
Although relabeling m 3 < m 1 < m 2 as m ′ 1 < m ′ 2 < m ′ 3 does not add any new physical content, we stress that it represents a phenomenologically alternative and interesting way to describe the neutrino mass spectrum and the lepton flavor mixing pattern, especially when they are compared with (and even unified with) the "normal" quark mass spectra and the CKM quark flavor mixing matrix. That is why we intend to look into such an option in some detail, and hope that this kind of study may help make the underlying physics more where "T" denotes the transpose, M ν ≡ Diag{m 1 , m 2 , m 3 } and M ′ ν ≡ Diag{m ′ 1 , m ′ 2 , m ′ 3 } = Diag{m 3 , m 1 , m 2 }. Eq. (7) tells us that M ν keeps invariant under the neutrino mass relabeling.
This observation is of course expectable because any physics must be unchanged by reordering the neutrino mass spectrum itself. It implies that a possible flavor symmetry of M ν will give rise to the same constraints on the patterns of U and U ′ .
To see the above point of view in a more transparent way and illustrate the relevant physics, let us impose the flavor transformation on the neutrino mass term L mass in Eq. (5) and require the latter to be invariant. It is easy to see that the invariance of L mass under such a µ-τ reflection transformation [11] cannot hold unless the elements of M ν satisfy the four conditions [12] In view of Eq. (7), it is easy to show that the texture of M ν constrained by Eq. (9) can also be diagonalized by Because of U ′ = US −1 , the locations of φ and ϕ in U ′ are different from those in U. We conclude that the µ-τ reflection symmetry of M ν leads to both |U µi | = |U τ i | and |U ′ µi | = |U ′ τ i | (for i = 1, 2, 3), although U and U ′ correspond to the different neutrino mass orderings. 2 In this parametrization c ij ≡ cos θ ij and s ij ≡ sin θ ij (for ij = 12, 13, 23) are defined, and all the three mixing angles are arranged to lie in the first quadrant. As for the CP-violating phases, δ varies from 0 to 2π, but φ and ϕ are only allowed to vary between 0 and π. Note that the sign convention of U µi (for i = 1, 2, 3) in Eq. (10) is different from that advocated by the Particle Data Group [4], because the latter will lead to . In Eq. (12) the flavor mixing matrix U ′ is required to take the same new sign convention as U . The generic relationships between the flavor mixing parameters (θ 12 , θ 13 , θ 23 , δ) in U and their counterparts in U ′ are given by where t ′ ij ≡ tan θ ′ ij , c δ ≡ cos δ, s δ ≡ sin δ and s ′ δ ≡ sin δ ′ . It is easy to use Eq. (14) to check that θ ′ 23 = θ 23 = π/4 and δ ′ = δ = π/2 or 3π/2 hold in the µ-τ reflection symmetry limit. In Table 1 we list the latest global-fit results of six neutrino oscillation parameters [6]. It is obvious that the best-fit value of δ is close to δ = 3π/2 as indicated by the µ-τ reflection symmetry of M ν . Hence we shall only take δ = 3π/2 in the remaining part of this section when discussing the µ-τ reflection symmetry limit. With the help of  To be more specific, the four flavor mixing parameters in the standard parametrization of U or U ′ read as together with We see that the value of θ ′ 12 is more than two times larger than that of θ 12 , and it is even not far away from 90 • . In addition, θ ′ 13 is about four times larger than θ 13 . But θ ′ 23 and θ 23 are roughly comparable in magnitude, so are δ ′ and δ. Figure 3 gives a more intuitive comparison between the results of θ ij and θ ′ ij obtained in Eq. (17), where the fractional error bar of each flavor mixing angle is estimated from the difference between the upper and lower bounds of its 3σ range divided by its central value as listed in Table 1, and the corresponding errors translate from θ ij to θ ′ ij via Eq. (14).
How about the parameters of flavor mixing and CP violation in a different parametrization of U and U ′ ? To answer this question, we list all the nine angle-phase parametrizations of U and U ′ [13] in Table 2 and calculate the best-fit values of the corresponding four flavor mixing parameters 3 . Some comments are in order.
• The benchmark parametrization of U or U ′ is P3, which is actually consistent with the standard one given in Eq. (10) Table 1, one may first determine the corresponding values of θ ′ 12 , θ ′ 13 , θ ′ 23 and δ ′ of U ′ in the form of P3, and then translate the relevant results into the other eight parametrizations of U and U ′ as shown in Table 2.
• Among the nine parametrizations of U and U ′ , only P1 [14] allows all the three mixing angles of U ′ to be comparable in magnitude and smaller than 60 • for the given inputs.
In comparison, the outputs of four flavor mixing parameters of U ′ in P2 (the Kobayashi-Maskawa parametrization [3]) are quite similar to those in P3, with θ ′ 12 ∼ 80 • . Another interesting parametrization of U ′ , P5 [15], contains θ ′ 12 70 • . One can see that the parametrizations of U ′ under discussion may involve one or two large mixing angles in most cases, as a straightforward consequence of the reordering of the inverted neutrino When the µ-τ reflection symmetry is taken into account, as illustrated in Table 3, we find that only P2, P3 and P7 are suitable in this connection because they simply lead us to θ 23 = θ ′ 23 = π/4 and δ = δ ′ = π/2 or 3π/2 as well as φ = 0 or π/2 and ϕ = 0 or π/2. Although |U µi | = |U τ i | and |U ′ µi | = |U ′ τ i | (for i = 1, 2, 3) also hold for the other six parametrizations of U and U ′ in the µ-τ reflection symmetry limit, the structures of these parametrizations make themselves less interesting in describing the phenomenology of neutrino oscillations.
At this point let us briefly comment on the so-called quark-lepton complementarity relations θ 12 + θ q 12 = π/4 [17] and θ 23 ± θ q 23 = π/4 [18] in the standard parametrization of the PMNS matrix U and the CKM matrix V . Given θ q 12 = 13.023 • ± 0.038 • , θ q 13 = 0.201 +0.009 • −0.008 • , θ q 23 = 2.361 +0.063 • −0.028 • and δ q = 69.21 +2.55 • −4.59 • extracted from current experimental data on quark flavor mixing and CP violation [4], such relations seem acceptable as a phenomenological conjecture 4 . One may follow a similar way to conjecture the relations like θ ′ 12 + θ q 12 = π/2 and θ ′ 23 ± θ q 23 = π/4 [19] when reordering the inverted neutrino mass hierarchy and taking account of the standard parametrization of U ′ , but they seem unlikely to shed light on the underlying dynamics of flavor mixing. 3 Because the Majorana phases φ and ϕ are completely insensitive to neutrino oscillations and completely unconstrained, here we only consider the Dirac CP-violating phase δ or δ ′ in our examples. 4 Note that U and V are associated respectively with W − and W + , and hence it is more appropriate to compare between U and V † in some sense [19]. But V itself is approximately symmetric, so the three mixing angles of V † are equal to those of V to a good degree of accuracy.

The µ-τ symmetry breaking
At the level of the PMNS matrix U or U ′ , a deviation of θ 23 or θ ′ 23 from π/4 is the socalled octant problem which is one of the important concerns in current neutrino oscillation experiments. On the other hand, a departure of δ or δ ′ from 3π/2 can be referred to as the quadrant problem, provided the present best-fit value of δ is essentially true. Both issues actually measure the effects of µ-τ reflection symmetry breaking, and hence it makes sense to examine such effects with the help of Table 1.
At the level of the Majorana neutrino mass matrix M ν , the µ-τ reflection symmetry is equivalent to the four relations given in Eq. (9). That is why some authors [20] have used the imaginary parts of m ee and m µτ and the differences m eµ − m * eτ and m µµ − m * τ τ to measure the symmetry breaking effects. But these four quantities depend on the phase convention of the charged-lepton fields, and hence they are not fully physical.
Here we illustrate the effects of µ-τ reflection symmetry breaking in a somewhat different way. We first calculate the profiles of | m αβ | (for α, β = e, µ, τ ) versus the lightest neutrino mass (either m 1 in the normal ordering or m 3 in the inverted ordering) by inputting the 3σ ranges of the relevant neutrino oscillation parameters listed in Table 1. For each of the six profiles, we figure out the much narrower areas fixed by the µ-τ reflection symmetry conditions θ 23 = π/4, δ = π/2 or 3π/2, φ = 0 or π/2 and ϕ = 0 or π/2 in the standard parametrization of U. Note that the case of δ = π/2 is equivalent to that of δ = 3π/2, because | m αβ | only contains cos δ. Therefore, we are left with four different situations with (φ, ϕ) = (0, 0), (π/2, 0), (0, π/2) and (π/2, π/2), as shown in Figure 4 for the normal neutrino mass ordering or in Figure 5 for the inverted mass ordering. Some discussions are in order.
• In either Figure 4 or Figure 5, the µ-τ reflection symmetry reflected by the relationships | m eµ | = | m eτ | and | m µµ | = | m τ τ | can clearly be seen. Given m 1 0.1 eV for the normal mass ordering or m 3 0.1 eV for the inverted mass ordering, a reasonable upper bound extracted from current cosmological data [21], we find that either | m µτ | in Figure 4 or | m ee | in Figure 5 can be most stringently constrained. Hence the effect of µ-τ reflection symmetry breaking for either of them is relatively modest even at the 3σ level. In comparison, the symmetry breaking effects are completely unconstrained at the 3σ level for | m eµ |, | m µµ |, | m eτ | and | m τ τ | in the inverted mass ordering case, as shown in Figure 5. For m 1 2 × 10 −2 eV in the normal mass ordering case, Figure 4 shows that | m µµ | and | m τ τ | are well constrained (around a few ×10 −2 eV), and the corresponding µ-τ reflection symmetry breaking effects are quite modest.
• The size of | m ee | determines the rates of the neutrinoless double-beta (0ν2β) decays for some even-even nuclei, which are the only feasible way at present to probe the Majorana nature of massive neutrinos [22]. It is obvious that the inverted mass ordering is more favorable for this purpose, because even the lower limit of | m ee | is above 0.01 eV, which is experimentally accessible in the foreseeable future [23]. As for the normal neutrino mass ordering, what interests us most is the possibility of significant cancellation among the three components of m ee even in the µ-τ reflection symmetry case (the green and red regions in the profile of | m ee |, as shown in Figure 4). To understand this observation, we follow Ref. [24] to express | m ee | in the following way in terms of two Majorana phases ρ and σ: where ρ = 2 (φ − ϕ) and σ = −2 (δ + ϕ) in the phase notations defined in Eq. (10). As pointed out in Ref. [24], a necessary condition for | m ee | → 0 is ρ = ±π, which are equivalent to (φ, ϕ) = (π/2, 0) (the green region) and (φ, ϕ) = (0, π/2) (the red region) in Figure 4, respectively. That is to say, among the four possible combinations of the two Majorana phases φ and ϕ in the µ-τ reflection symmetry limit, two of them are likely to make | m ee | fall into a well (i.e., make the rate of a 0ν2β decay too small to be observable) if the value of m 1 happens to lie in a "wrong" region 5 .
Of course, it is very difficult to pin down the effects of µ-τ reflection symmetry breaking at the neutrino mass matrix level unless the octant issue of θ 23 is experimentally resolved, the absolute neutrino mass scale is fixed or at least further constrained, and the three CP-violating phases are determined to an acceptable degree of accuracy.
But even the preliminary results obtained in Figures 4 and 5 can tell us something useful.
For example, one may draw the conclusion that the well-known Fritzsch texture [25], must not be applicable to the inverted neutrino mass ordering even if the latter is relabeled to be normal. The reason is simply that m ee is not only nonzero but also sizeable in this case, as shown in Figure 5. Another interesting observation is that there is strong tension between M (F) ν and current experimental data at the 3σ level even in the normal mass ordering case, as Figure 4 shows that the matrix elements m ee and m µµ cannot simultaneously vanish 6 .
Another instructive example is the following two-zero texture of M ν , which is usually referred to as "Pattern C" and favored with the inverted neutrino mass ordering [27,28]: A comparison of Eq. (21) with Figure 5 tells us that such a special texture of M ν remains compatible with current neutrino oscillation data at the 3σ level, but it does not respect 5 As for the inverted neutrino mass ordering, | m µτ | → 0 is also possible in the µ-τ reflection symmetry limit with φ = π/2 and ϕ = 0 (the green region in Figure 5). 6 This result is in conflict with the previous studies based on very preliminary neutrino oscillation data [26].
the µ-τ reflection symmetry which definitely forbids vanishing or very small (generously say, 10 −3 eV) values of m µµ and m τ τ .
Therefore, we expect that our model-independent results shown in Figures 4 and 5 can help for the model-building exercises in connection to the neutrino mass spectrum, flavor mixing and CP violation, no matter whether one follows the flavor symmetry approach or the texture-zero approach, or a mixture between them.

Summary
We have asked ourselves what to do if the neutrino mass spectrum is finally found to be inverted. A straightforward and phenomenologically meaningful choice is certainly to relabel such that the latter is as normal as the mass spectra of those charged leptons and quarks of the same electrical charges. But in this case the columns of the PMNS matrix U must be reordered accordingly, and the resulting pattern U ′ turns out to involve an especially large mixing angle in the standard parametrization. We have examined the other angle-phase parametrizations of U and U ′ , and reached a similar observation. Given the fact that the Majorana neutrino mass matrix M ν keeps unchanged in such a mass relabeling exercise, we have considered the intriguing µ-τ reflection symmetry to illustrate the texture of M ν and the effects of its symmetry breaking at the 3σ level by using current neutrino oscillation data. Our model-independent results are expected to be helpful for building specific neutrino mass models.
What is really behind the normal or inverted neutrino mass ordering remains an open question, and whether there is a definite correlation between the fermion mass spectra and flavor mixing patterns is also a big puzzle. The underlying flavor theory, which might be based on a certain flavor symmetry and its spontaneous or explicit breaking mechanism, may finally answer the above questions. To approach such a theory, we are now following a phenomenological way from the bottom up. Much more efforts in this connection are needed.
We would like to thank Y.  The best-fit values of three flavor mixing angles and the Dirac CP-violating phase for the inverted neutrino mass ordering in each of the nine parametrizations of U and U ′ , where s θ ij ≡ sin θ ij , s ϑ ij ≡ sin ϑ ij , c θ ij ≡ cos θ ij and c ϑ ij ≡ cos ϑ ij (for ij = 12, 13, 23). Table 3: The best-fit values of θ 12 and θ 13 together with θ 23 = π/4 and δ = 3π/2 for P3 in the µ-τ reflection symmetry limit [16] and in the inverted neutrino mass ordering, and their variations for the other eight parametrizations in the same case. Figure 4: The profiles of | m αβ | versus the lightest neutrino mass m 1 for the normal neutrino mass ordering, where the 3σ ranges of six neutrino oscillation parameters [6] have been input.
The pink, red, green and blue regions are fixed by the µ-τ reflection symmetry with θ 23 = π/4 and the special values of δ, φ and ϕ listed on top of the figure.