_{23}and third quadrant of

Supported by the National Natural Science Foundation of China (11775231, 11775232)

_{23}and third quadrant of

^{3}and published under licence by Chinese Physical Society and the Institute of High Energy Physics of the Chinese Academy of Sciences and the Institute of Modern Physics of the Chinese Academy of Sciences and IOP Publishing Ltd

The recent global analysis of three-flavor neutrino oscillation data indicates that the normal neutrino mass ordering is favored over the inverted one at the 3_{23} and the Dirac CP-violating phase _{1} and the MSSM parameter tan

Article funded by SCOAP^{3}

The striking phenomena of solar, atmospheric, reactor and accelerator neutrino oscillations have all been observed in the past twenty years [

Because the oscillation experiments are insensitive to the Dirac or Majorana nature of massive neutrinos, one may describe the link between the three known neutrinos (_{e}_{μ}_{τ}_{1}, _{2}, and _{3}) in terms of a 3 × 3 unitary matrix, namely the Pontecorvo–Maki–Nakagawa–Sakata (PMNS) matrix [_{ij}_{ij}_{ij}_{ij}_{1} ˂ _{2} ˂ _{3}) is favored over the inverted one (_{3} ˂ _{1} ˂ _{2}) at the 3^{4)}. Furthermore, the best-fit value of the largest neutrino mixing angle _{23} is slightly larger than 45° (i.e., located in the higher octant), and the best-fit value of the Dirac phase _{23} = 45° and _{ν}^{1)}, a simple flavor symmetry that assures that _{ν}_{μτ} ~ 10^{14} GeV, then it will be broken at the electroweak scale Λ_{EW} ~ 10^{2} GeV owing the RGE running effects, leading to deviations of (_{23}, _{23} and _{23}, and the strength of leptonic CP violation.

Given the fact that a credible global analysis of relevant experimental data often points to the truth in particle physics^{2)}, it is necessary for us to take the latest global-fit results of neutrino oscillations seriously, and to investigate their implications independently of any details of model building. In this study, we show that the normal neutrino mass ordering, the slightly higher octant of _{23}, and the possible location of _{μτ} ~ 10^{14} GeV down to Λ_{EW} ~ 10^{2} GeV in the minimal supersymmetric standard model (MSSM). This kind of correlation will soon be tested by more accurate experimental data.

Although the same topic has been discussed previously, the present study differs from previous works at least in the following aspects:

Based on more reliable experimental data, especially the 3_{1} and the MSSM parameter tan_{23} by assuming that _{13} = 0. This is an approximation that has now been excluded. Such work also did not consider all four distinct cases of the Majorana phases in the

While all previous studies have simply assumed some special values of _{1} and tan_{23} and the quadrant of

Our statistical analysis shows that currently the best-fit points of _{23} and _{μτ} down to Λ_{EW}, but this simple flavor symmetry itself is in slight disagreement (at the 1

The upcoming precision measurement of _{23} and an experimental determination of _{1}, the MSSM parameter tan

_{1}is extensively explored and constrained with the help of current experimental data. Section

Let us assume that the tiny masses of the three known neutrinos originate from a viable seesaw mechanism at a superhigh energy scale Λ_{μτ} ~ 10^{14} GeV. Without loss of generality, we choose the basis in which the mass eigenstates of the three charged leptons are identical with their flavor eigenstates. In this case, only the neutrino sector is responsible for lepton flavor mixing and CP violation. If the effective Majorana neutrino mass term is invariant under the charge-conjugation transformations _{l}V P_{ν}_{l}^{iϕe}, e^{iϕμ}, e^{iϕτ}} and _{ν}^{iρ}, e^{iσ}, 1} being the phase matrices, and _{ν}_{23} = 45°, _{e}_{μ}_{τ}

In the framework of the MSSM, the evolution of _{ν}_{μτ} down to Λ_{EW} through the one-loop RGE can be expressed as [_{l}_{e}_{μ}_{τ}_{μτ}), where _{EW} and Λ_{μτ}, _{1} and _{2} denote the gauge couplings, and _{t}_{α}_{e}_{μ}_{e}_{μ}_{τ} (left panel) and _{0} (right panel) versus the variables Λ_{μτ} and tan_{0} does not change significantly with different settings of Λ_{μτ} and tan_{τ} can change from 0.001 to 0.05. Note that shifting the energy scale is equivalent to altering tan_{μτ} = 10^{14} GeV and Λ_{μτ} = 10^{16} GeV are reasonably similar in magnitude. If we shift the energy scale from Λ_{μτ} = 10^{14} GeV to Λ_{μτ} = 10^{9} GeV, then Δ_{τ} will lie in the range (0.001,0.03) instead of (0.001,0.05). In the following numerical calculations, we shall fix Λ_{μτ} ~ 10^{14} GeV as the

(color online) Possible values of Δ_{τ} (left panel) and _{0} (right panel) versus the _{μτ} and MSSM parameter tan

One may diagonalize the neutrino mass matrix at Λ_{EW}, and then obtain the mass eigenvalues (_{1}, _{2}, _{3}), flavor mixing angles (_{12}, _{13}, _{23}), and CP-violating phases (_{ij}_{ij}_{EW}) − _{ij}_{μτ}) (for _{EW}) − _{μτ}), Δ_{EW}) − _{μτ}), and Δ_{EW}) − _{μτ}) to measure the strengths of the RGE-induced corrections to the parameters of _{EW} are found to be
_{12} and _{13} take their values at Λ_{EW}. Unless otherwise specified, the nine physical flavor parameters appearing in the subsequent text and equations are all those at Λ_{EW}. In a reasonable analytical approximation, we can also arrive at^{1)}
_{EW} and Λ_{μτ}, and
_{EW} and Λ_{μτ}, where _{ρ}_{σ}_{μτ}, and the ratios _{ij}_{i}_{j}_{i}_{j}_{i}_{j}_{EW} (for _{23}(Λ_{μτ}) = 45° and _{μτ}) = 270° have also been applied.

Although the Majorana nature of massive neutrinos is well motivated from a theoretical viewpoint, it is also interesting to consider the possibility of a pure Dirac mass term for the three known neutrinos, and combine this with a certain flavor symmetry that can be realized at a superhigh energy scale Λ_{μτ} [_{eL} ↔ (_{eL})^{c}, _{μL} ↔ (_{τL})^{c}, and _{τL} ↔ (_{μL})^{c} for the left-handed neutrino fields and _{eR} → (_{eR})^{c}, _{μR} ↔ (_{τR})^{c}, and _{τR} ↔ (_{μR})^{c} for the right-handed neutrino fields. The resultant Dirac neutrino mass matrix is
_{ν}_{l}V_{l}^{iϕe}, e^{iϕμ}, e^{iϕτ}} being an unphysical phase matrix and _{ν}_{23} = 45°, _{e}_{μ}_{τ}_{μτ}. Given the global-fit preference for sin_{μτ}) = 270°.

In the MSSM, the evolution of the Dirac neutrino mass matrix _{ν}_{μτ} down to Λ_{EW} via the one-loop RGE can be described as [_{0} and _{l}_{ν}_{EW}) will yield the seven physical flavor parameters at Λ_{EW}. Utilizing the same notations as in the Majorana case, we summarize our approximate analytical results as follows:
_{EW} and Λ_{μτ}, and
_{EW} and Λ_{μτ}. In obtaining Eqs. (_{23}(Λ_{μτ}) = 45° and _{μτ}) = 270°.

The analytical approximations presented in Eqs. (

In the framework of the MSSM, we numerically run the RGEs from Λ_{μτ} ~ 10^{14} GeV down to Λ_{EW} ~ 10^{2} GeV by taking into account the initial conditions _{23} = 45° and _{1} at Λ_{EW}, the other relevant neutrino oscillation parameters, such as {sin^{2}_{12}, sin^{2}_{13}, _{μτ} are scanned over suitably wide ranges with the help of the MultiNest program [^{1)}. For each scan, the neutrino flavor parameters at Λ_{EW} are obtained, and these are immediately compared with their global-fit values by minimizing
_{i}_{EW} produced from the RGE evolution, the _{i}^{2)} and adopt the best-fit values and 1_{1} is allowed to take values in the range [0, 0.1] eV, and the MSSM parameter tan^{3)}. It should be pointed out that our numerical results are independent of the analytical approximations obtained in the previous section, but the latter will be helpful for understanding some salient features of the former.

The strategy of our numerical analysis is rather straightforward. Let us first examine how significantly _{23} and _{EW} can deviate from their initial values at Λ_{μτ}, incorporating the recent global-fit results. To this end, we only need to take into account the global-fit information concerning the parameter set ^{2}_{12}, sin^{2}_{13}, _{23} to the higher octant, and in most cases lead _{23} and _{23} and

For each given value of _{1} or tan_{EW}, we obtain the associated _{1} for Case C as an example. It is obvious that _{1} ∈ [0.05, 0.08] eV the value of _{μτ} cannot be reached if one runs the RGEs inversely, starting from the best-fit points of the six neutrino oscillation parameters at Λ_{EW}. This observation will also be true even if one allows _{23} and _{EW}. The reason for this should be ascribed to the nontrivial differential structures of the RGEs [_{EW} up to Λ_{μτ}. The squared distance to the _{μτ} is defined as
_{i}_{EW} up to Λ_{μτ} with {sin^{2}_{12}, sin^{2}_{13}, _{EW}. As shown in Fig. ^{2} bump the same order of magnitude as that of _{μτ} down to Λ_{EW}.

(color online) The behavior of _{1} for Case C, (i.e., _{μτ}), where tan

Figure _{23} at Λ_{EW} for different values of _{1} and tan^{2} = 0, where the four possible initial options for (_{μτ} have been considered. Note that the boundary conditions for the RGEs include both the initial values of {_{23}, _{μτ} and the experimental constraints on {sin^{2}_{12}, sin^{2}_{13}, _{EW}, which are all specified in our numerical calculations. In this case, one should keep in mind that the low- and high-scale boundary requirements are likely to be sufficiently strong that the RGEs do not yield a realistic solution, illustrated by the gray-gap regions in Fig.

The gray-gap regions in Cases C and D are a result of the _{23} may run to almost 75° for _{1} ≃ 0.1 eV and tan_{23}, which has not yet been included in our analysis. In contrast, _{23} is not sensitive to the RGE corrections in Case B, and it maximally changes by only approximately 1°. For Cases C and D, if one conservatively requires that _{1} ˂ 0.07 eV from the present cosmological bound [_{23} to run to 46.6° and 50°, respectively. The best-fit value _{23} ≃ 48° [

The RGE correction to _{23} illustrated in Fig. _{23} is positive, because Δ_{τ} and _{31} ≃ _{32} are all positive for the normal neutrino mass ordering. The factor Δ_{τ} is essentially proportional to tan^{2}_{23} always increases with tan^{2}_{23} on the neutrino mass _{1} is different for the four options of _{μτ}. For example, _{1} for _{μτ}) = 0°, but is proportional to 1/_{1} when _{μτ}) = 90°. In the region of small _{1} and tan_{23} is proportional to _{1} for Cases A, C, and D, but is inversely proportional to _{1} in Case B with _{ρ}_{σ}

(color online) The allowed region of _{23} at Λ_{EW} owing to the RGE-induced _{23}, and the blue one is compatible with the best-fit result of _{23} obtained in Ref. [

In Fig. _{EW}. Note that for each point in the _{1}-tan_{23} are determined concurrently. Some remarks are in order.

The RGE-induced corrections to _{23} (including its best-fit value) can easily be reached in Case A, it is impossible to approach the best-fit value of _{23},_{1} and tan

Similar to the case of _{23}, the radiative correction to ^{2}_{1} in Eq. (_{23} in Eq. (_{13} and the other suppressed by sin_{13}, but the latter can become dominant in some cases. In Case A, the first term ∝ 1/sin_{13} is positive and dominant when the neutrino mass _{1} is relatively small, while the second term ∝ sin_{13} is negative, and will gradually dominate when the value of _{1} increases. These analytical features can explain the numerical evolution behavior of _{ρ}_{σ}_{1}. Note that the first term of Δ_{1}. Hence, the RGE-induced corrections to

(color online) The allowed region of _{EW} owing to the RGE-induced

To see the correlation between _{23} and _{EW}, let us marginalize _{1} and tan_{1} ∈ [0,0.1] eV and tan_{23},_{EW}, which is marked by the red circled cross in the plot, lies on the dashed contour. This means that _{23}(Λ_{EW}) = 45° and _{EW}) = 270° are statistically unfavored at the 1_{23}-^{2}_{12}, sin^{2}_{13}, _{23} and _{23} ≃ 50° in Fig. _{23} and

(color online) The correlation between _{23} and _{EW} compared with the recent global-fit results (abbreviated as “CLMP”) [_{1} and tan_{23},_{23} and ^{2}_{12}, sin^{2}_{13}, _{EW} take their best-fit values, and the green region is allowed when these four observables deviate from their best-fit values by a 3^{2} = 11.83 for two degrees of freedom).

To numerically verify the compatibility between our _{EW}, now we include the global-fit information on _{23} and ^{2} is formed with the parameter set _{23} and _{μτ} by marginalizing the relevant quantities over tan_{1}. For the special point (_{23},_{EW}, the corresponding _{μτ} is most favorable. Even given the Planck limit on the sum of the neutrino masses Σ ≡ _{1} + _{2} + _{3} ˂ 0.23 eV at the 95% confidence level [

(color online) The minimal _{23},_{EW}.

Because there is only a single CP-violating phase in the Dirac case, it is considerably easier to perform a numerical analysis of the parameter space, which is constrained by both the RGE-induced _{23} and _{EW}, and their intimate correlation is illustrated in Fig. _{23} and Δ_{23}, _{1} and tan_{23} and _{23},

(color online) The allowed regions of _{23} (left panel) and _{EW} owing to the RGE-induced _{23} and _{23} or

(color online) The correlation of the broken values of (_{23},

In neutrino physics, it is usually necessary (and popular) to introduce some heavy degrees of freedom and certain flavor symmetries at a superhigh energy scale in order to explain the tiny masses of the three known neutrinos and the striking pattern of lepton flavor mixing observed at low energies. In this case, it is also necessary to employ the RGEs as a powerful tool to bridge the gap between these two considerably different energy scales. Such RGE-induced quantum corrections may naturally break the given flavor symmetry, thus leading to some phenomenologically interesting consequences, including a possible correlation between the neutrino mass ordering and flavor mixing parameters.

In this work, we have considered the intriguing _{23} and the possible location of _{μτ} ~ 10^{14} GeV down to the electroweak scale Λ_{EW} ~ 10^{2} GeV in the MSSM. Unlike previous attempts along these lines, our study represents the first numerical exploration of the complete parameter space in both the Majorana and Dirac cases, by allowing the smallest neutrino mass _{1} and the MSSM parameter tan_{23} and

Of course, some of our main observations are subject to the MSSM itself, and the current best-fit values of _{23} and _{23} via the RGE-induced _{23} from Λ_{μτ} down to Λ_{EW} seems to be “wrong” if one takes today’s best-fit result _{23} ˃ 45° seriously in the normal mass ordering case; and (c) the SM itself may suffer from the vacuum-stability problem, as the energy scale is above 10^{10} GeV [_{23}, and the quadrant of _{23} and

On the other hand, we admit that the best-fit values of _{23} and _{1} and tan_{23} and the third quadrant of _{23}, and (or) another quadrant of

We admit that the inverted neutrino mass ordering is currently still allowed at the 2

As in most literature, in this case the so-called _{μ}_{τ}

One successful example of this kind was the global-fit “prediction” for an unsuppressed value of _{13}, made in 2008 [_{13} was reported in 2012 [

Note that our analytical results are not exactly the same as those obtained in Ref. [

In the normal neutrino mass ordering case, one may therefore express ^{2} and ^{3} in terms of _{1} as

This is not only our phenomenological preference but also a recent 3_{1} ˂ _{2} ˂ _{3}.

Note that tan