PLB140477

140477

S0370-2693(26)00330-810.1016/j.physletb.2026.140477The Authors

Fig. 1

The predicted probability distributions of the leptonic Dirac CP phase according to the residual Z2s (red) or Z‾2s (blue) symmetries with inputs from the current NuFIT 6.0 global fit plus the JUNO first data release. In addition, the CP phase probability distributions from the NuFIT 6.0 global fit (purple dashed) and the T2K plus NOvA joint analysis (cyan dashed) are also shown for comparison. Both normal (NO) and inverted (IO) orderings have been shown in the upper and lower panels, respectively.

Fig. 1 dummy alt text

Fig. 2

Comparison between the theoretical prediction of the residual Z2s (red) or Z‾2s (blue) symmetries and the latest NuFIT 6.0 results (green) for the two-dimensional correlated probability distributions between the atmospheric angle θa and the leptonic Dirac CP phase δD. The upper panel is for NO while the lower one is for IO.

Fig. 2 dummy alt text

Table 1

Bayesian evidence P(D|M) and Bayes Factors BF ≡P(D|Z2s)/P(D|Z‾2s) with inputs from the NuFIT 6.0 (T2K-NOvA joint analysis) and the JUNO first data. For the analysis with both orderings, we are assuming equal probability for normal and inverted ordering.

Table 1 dummy alt text

SymmetryBF (1D)BF (2D)NOZ2s0.42 (0.59)0.30Z‾2sIOZ2s2.11 (2.02)2.40Z‾2sNO & IOZ2s1.18 (1.13)1.35Z‾2sLetterCP prediction from residual Z2s and Z‾2s symmetries with the latest data

CP prediction from residual Z2s and Z‾2s symmetries with the latest data

Shao-Feng

gesf@sjtu.edu.cn

Chui-Fan

Kong

kongcf@ibs.re.kr

João Paulo

Pinheiro

joaopaulo.pinheiro@fqa.ub.edu

aSchool of Physics & Astronomy, State Key Laboratory of Dark Matter Physics, Tsung-Dao Lee Institute, Shanghai Jiao Tong University, China School of Physics & Astronomy State Key Laboratory of Dark Matter Physics Tsung-Dao Lee Institute Shanghai Jiao Tong University China

School of Physics & Astronomy, State Key Laboratory of Dark Matter Physics, Tsung-Dao Lee Institute, Shanghai Jiao Tong University, China

bKey Laboratory for Particle Astrophysics & Cosmology (MOE) & Shanghai Key Laboratory for Particle Physics & Cosmology, Shanghai Jiao Tong University, Shanghai, 200240, China Key Laboratory for Particle Astrophysics & Cosmology (MOE) & Shanghai Key Laboratory for Particle Physics & Cosmology Shanghai Jiao Tong University Shanghai

200240

China

Key Laboratory for Particle Astrophysics & Cosmology (MOE) & Shanghai Key Laboratory for Particle Physics & Cosmology, Shanghai Jiao Tong University, Shanghai, 200240, China

cCenter for Theoretical Physics of the Universe (CTPU), Particle Theory and Cosmology Group (PTC), Institute for Basic Science, Daejeon, 34126, Republic of Korea Center for Theoretical Physics of the Universe (CTPU) Particle Theory and Cosmology Group (PTC) Institute for Basic Science Daejeon

34126

Republic of Korea

Center for Theoretical Physics of the Universe (CTPU), Particle Theory and Cosmology Group (PTC), Institute for Basic Science, Daejeon, 34126, Republic of Korea

Editor: Prof Ryuichiro Kitano

Abstract

The JUNO first data and the recent neutrino global fit results are implemented in the sum rule from the residual Z2s and Z‾2s symmetries to make prediction of the leptonic Dirac CP phase δD. Without involving model parameters, the probability distribution of δD can be readily obtained from the experimental measurements of the three mixing angles. We then confront the theoretical predictions with the global fit results for the CP phase as well as the T2K and NOvA joint analysis for their CP measurements to give the data preference of the two residual symmetries with Bayes factor for both normal and inverted orderings. We further extend our analysis to a two-dimensional probability distribution to fully explore the correlation between the CP phase δD and the atmospheric angle θa ≡ θ23.

Keywords

Neutrino CPResidual symmetryNeutrino oscillation

Data availability

Data will be made available on request.

Introduction

Neutrino oscillation is the first new physics beyond the Standard Model of particle physics with various solid experimental data [1]. It provides not just a key to understanding the possible new physics world beyond our current knowledge [2–5], but also a first observation of quantum interference phenomena at macroscopic lengths spanning from O(1) km at reactor experiments such as Daya Bay to O(105) km for solar neutrino transition.11

Note that the 2025 Nobel Prize in Physics awards quantum phenomena at the centimeter scale.

With the Daya Bay [6], RENO [7], and Double Chooz [8] experiments establishing a nonzero θr( ≡ θ13) mixing angle, neutrino physics has entered a precision era [9]. Especially, this allows possible measurement of the leptonic Dirac CP phase δD [10,11] that may hold the key to understanding why our Universe is made of matter but almost no anti-matter [12,13].The mixing parameters will be measured with high precision at the new-generation neutrino experiments. Especially, the JUNO experiment can provide sub-percentage precision for θs( ≡ θ12) and Δms2 (≡Δm212) [14,15] with full data collection. This would allow further test of those flavor symmetries and models that can predict the neutrino mixing pattern.A flavor symmetry is typically imposed on the fundamental Lagrangian of an ultra-violet (UV) complete theory as starting point [16–22]. Being an UV complete model, it should satisfy the electroweak gauge symmetries SU(2)L × U(1)Y with the left-handed neutrino νℓL and its charged lepton counterpart ℓL in the same SU(2)L doublet (νℓL, ℓL)T. The flavor symmetry should be imposed on such doublet to respect the gauge symmetries. Then, the flavor symmetry has to be broken in order to allow the left-handed neutrinos and charged leptons to develop different mixing matrices Uν and Uℓ, respectively, such that the physical PMNS mixing matrix UPMNS≡Uℓ†Uν can be nontrivial [23]. Such flavor symmetry breaking should happen at the same time as the electroweak symmetry breaking. So if there is any flavor symmetry that really dictates the neutrino mixing pattern, it has to be the residual symmetry [24] that survives the symmetry breaking processes.After symmetry breaking, a residual symmetry should apply to the neutrino mass term whose diagonalization gives the mixing pattern. It turns out a direct connection can be established not just between the symmetry transformation Gν and the neutrino mass matrix Mν but actually between Gν and the mixing matrix Uν [24–27]. Then a correlation among the mixing parameters (including three mixing angles and one leptonic Dirac CP phase) can be established without involving any model parameters or the neutrino mass eigenvalues [28–30]. Study on such correlation was later further developed under the name of Sum Rule.In this letter, we first summarize the major features and unique predictions of the residual Z2s or Z‾2s symmetries for Majorana neutrinos. Since the Dirac CP phase has not been fully established, we emphasize its predicted values in terms of the already measured three mixing angles [29–31]. Especially, the JUNO experiment provides very precise measurement of the solar angle θs [32]. We show the current prediction of δD has been significantly improved and make an explicit comparison with the latest measurement from the accelerator oscillation experiments [33].2

Residual symmetries and CP correlation with mixing angles

As mentioned above, a residual symmetry is the one that survives the electroweak symmetry breaking to allow different mixing matrices for neutrinos and charged leptons. Since neutrinos become massive after symmetry breaking, the transformation matrix Gν of a residual symmetry directly applies to the mass matrix Mν,(1)GνTMνGν=Mν.For simplicity, we have chosen the mass basis of charged leptons such that the PMNS matrix comes from the neutrino side alone. Note that the above symmetry transformation invariance of the mass matrix with a transposed GνT applies for Majorana neutrinos.The direct connection between a symmetry transformation matrix and the mass matrix is the essential feature of residual symmetries. If a symmetry is really the one that dictates the mixing pattern, it has to survive through the mass generation process and directly restricts the resulting mass matrix without interference from other factors such as Yukawa couplings and the vacuum expectation values.The mass matrix Mν of Majorana neutrinos in the flavor basis can be diagonalized as UνTMνUν=Dν where Dν ≡ diag{m1, m2, m3} is the diagonal mass matrix in the mass basis, or equivalently, Mν=Uν*DνUν†. Then one may replace the Mν in Eq. (1) to obtain GνTUν*DνUν†Gν=Uν*DνUν†. Although the neutrino mass eigenvalues mi are involved, it is possible to obtain a mass indepenent solution,(2)Uν†GνUν=dν,orGν=UνdνUν†.Under the transformation of dν, the diagonal mass matrix Dν should also be invariant, dνTDνdν=Dν which requires the dν to be diagonal with only 8 possibilities, dν ≡ diag{ ± 1,  ± 1,  ± 1}. The diagonal transformation dν is actually the diagonal representation of the same residual Z2 symmetries in the mass basis.It is interesting to observe that Eq. (2) is actually a direct connection between the residual transformation matrix Gν and the neutrino mixing matrix Uν. As indicated by the first equation therein, the mixing matrix Uν can be obtained by diagonalizing Gν. There is actually no need to first obtain the mass matrix Mν and then diagonalize it to determine the mixing matrix Uν. Instead of such two step procedure, the mixing pattern can be directly obtained from the residual symmetry transformation. The mixing pattern is really dictated by the residual symmetry without involving any other factors. If we take the relation in the other way around, the residual symmetry transformation matrix Gν can be reconstructed in terms of the mixing matrix Uν as explicitly shown by the second equation in Eq. (2). This is a very neat and direct connection between symmetry and observables.Of the 8 possibilities of dν only two are independent, dν(1)≡diag{−1,1,1} and dν(2)≡diag{1,−1,1}. Correspondingly, the two residual transformation matrices are,(3a)G1(k)=12+k22−k22k2kk2−2k2,(3b)G2(k)=12+k22−k22k2k−2k2−2, where k is a free parameter. Both G1 and G2 are generators of a Z2s or Z‾2s symmetry, respectively. Note that only one of these two Z2 symmetries can exist as residual symmetry, especially after the μ−τ symmetry [34,35] that corresponds to dν(3)≡diag{1,1,−1} and G3=G1G2 is broken.Although it seems like both G1(k) and G2(k) contain a model parameter k, they can predict unique connection between the Dirac CP phase δD and three mixing angles, which are the atmospheric angle θa ≡ θ23, the solar angle θs, and the reactor angle θr [29–31]:(4a)cosδD=(ss2−cs2sr2)(sa2−ca2)4casacssssr,(4b)cosδD=(ss2sr2−cs2)(sa2−ca2)4casacssssr, for Z2s and Z‾2s, respectively. Here sr,s,a ≡ sin θr,s,a and cr,s,a ≡ cos θr,s,a denote the sine and cosine functions of the mixing angles. Since the μ−τ symmetry dictates a vanishing reactor angle (θr = 0) and maximal atmospheric angle (θa=π/4), breaking it allows non-zero θa and ca2−sa2. The ratio between these two deviations is correlated with the leptonic Dirac CP phase δD.Although the transformation matrices G1(k) and G2(k) are functions of a model parameter k, the correlation above only involves physical observables in neutrino experiments. The property that a residual symmetry can establish connection among physical observables has a close analogy in the electroweak symmetry breaking. Although the SM SU(2)L × U(1)Y gauge symmetries are broken, the custodial symmetry predicts a correlation among the gauge boson masses (MZ and MW) and the weak mixing angle which are all physical observables. It is readily possible to use physical observations to justify such correlations. The residual symmetry for the neutrino mixing pattern has the same spirit as the custodial symmetry for the gauge mixing [36]. It is interesting to see that both cases have the concept of mixing pattern which is another similarity.3

CP prediction with JUNO first data

The current global-fit result [37] has reached percentage level for the mixing angle measurement and provided mild constraint on the Dirac CP phase. In addition, the JUNO reactor neutrino experiment just released their first data with better measurement of solar angle, ss2=0.3092±0.0087 [32]. With this in mind, we would like to see how the updated measurements affect the CP phase prediction of the residual symmetry sum rules. Especially, how the theoretical predictions of the Dirac CP phase distribution compare with its current measurement result.The probability distribution function of cos δD can be expressed by integrating over the mixing angle distribution probabilities [30,31],(5)dP(cosδD)dcosδD=∫δDpP(sr2)P(ss2)P(sa2)dsr2dss2dsa2,where δDp≡δ(cosδD−c¯D) is a δ-function with c¯D denoting the RHS of Eq. (4) to enforce the residual symmetry sum rule. The probability distribution function inside the integration P(sr2,ss2,sa2)≡P(sr2)P(ss2)P(sa2) represents the prior probability distributions of (sr2,ss2,sa2) extracted from data or global fits. Then, the probability distribution of the Dirac CP phase δD can be obtained with a simple transformation and Jacobian,(6)dP(δD)dδD=|sinδD|dP(cosδD)dcosδD.The prediction of the leptonic Dirac CP phase δD would sensitively depend on the input prior of mixing angles. In the presence of measurement uncertainties of mixing angles, the Dirac CP phase that predicted from the mixing angles is not a fixed value but follows a distribution. To calculate the predicted distribution of the Dirac CP phase from the mixing angles, we extract the NuFIT [37] one-dimensional distributions of the mixing angles.As indicated by Eq. (4), the two deviations (θr and ca2−sa2) from the tribimaximal mixing [38–40] are proportional to each other. With |cos δD| ≤ 1 being limited from above, the reactor angle θr needs to be nonzero which has been firmly established by Daya Bay [41], RENO [42], and Double-Chooz [43] with high precision. Otherwise, neither deviations can happen. A nonzero reactor angle θr is really the key for going beyond the tribimaximal mixing. With the precision measurements in the last 10 years, the uncertainty in sin 2θr has decreased from roughly 10% to around 2.5%.For these three mixing angles, we take their normalized distribution according to the corresponding χ2 function, P(sa,r,s2)∝exp(−χ2/2). Note that the χ2 functions of sr,s2 are almost symmetric and follow a parabola. Then, the probability distributions of sr,s2 can be described by a normalized Gaussian distribution function, P(sr,s2)≡exp[−(sr,s;BF2−sr,s2)2)/2σsr,s22]/2πσsr,s2. However, for the atmospheric angle, its distribution is asymmetric and has two local χ2 minima [37]. In this case, we calculate the sa2 distribution probabilities using its exact χ2 function.

Fig. 1

shows the predicted probability distribution function with the latest NuFit [37] plus the JUNO first data [32] (solid) for both Z2s (red) and Z‾2s (blue). In comparison with the previous results [31], the predicted CP distribution shrinks quite significantly. This is a manifestation of the fact that the experimental uncertainties on the mixing angles have improved quite a lot.

It is instructive to examine how the experimental uncertainty in θs propagates into the prediction of cos δD. Neglecting the tiny ss2 in the numerator, the sum rules in Eq. (4) are roughly modulated by the solar mixing angle as tan θs (for Z2s) and cotθs (for Z‾2s). For a small perturbation δθs in the solar angle, both tangent and cotangent propagate errors with the same amplification factor  ∼ 2δθs/sin (2θs), leading to a relative uncertainty in cos δD that is amplified by  ∼ 2/sin (2θs) compared to the relative uncertainty in θs itself. For the JUNO first measurement, this amplification factor is  ∼ 2.16. The improved precision from JUNO, reducing the relative uncertainty in θs from  ∼ 4% to  ∼ 2.16%, would improve the precision of the predicted cos δD by a factor of two from 8.6% to 4.7%.Comparing the solid curves that represent the updated prediction of the δD distribution with the previous results in 2013 [31], the updated predictions have two local peaks. Such a feature arises from the two local minima in the χ2 distribution function of sa2. To be more exact, the atmospheric angle θa has two local best-fit values in the lower (θa < π/4) or higher (θa > π/4) octants. According to Eq. (4), switching octant for θa would lead to a minus sign in the predicted cos θD and hence δD becomes π−δD. This explains why the two peaks mirror around the maximal CP point δD=±π/2. However, such mirror symmetry is not exact since the two octant solutions of θa have different probabilities or weights [9,37]. For NO, the two peaks have comparable heights while the IO case has a much larger disparity. But the previous results in 2013 are dominated by a single peak for the NO case, although the IO case has more sizable second peak. It seems that from the 2013 results to the updated, the situation on the atmospheric angle octant switches between NO and IO.Moreover, since the sum rule in Eq. (4) is determined by cos δD, the determination of the CP phase becomes symmetric between the [0, π] and [π, 2π] regions. As illustrated in Fig. 1, this dependence leads to an explicit degeneracy between δD and 2π−δD (or equivalently −δD). This cosine dependence arises because the residual Z2 symmetries in vacuum impose sum rules that fix δD via relations like |Uμivac|=|Uτivac|, making the solutions with opposite signs of δD [44] indistinguishable. However, the νμ → νe transition probability at long-baseline experiments is sensitivity to sin δD and can break this degeneracy. In realistic scenarios where neutrinos propagate through matter, this degeneracy is further lifted as the matter potential breaks the underlying vacuum symmetry. Long-baseline experiments such as DUNE [45] and T2HK [46] will exploit this matter-induced breaking of the vacuum symmetry to resolve the degeneracy in δD, allowing symmetry models to be tested against a unique determination of the CP-violating phase [47,48].In addition, the major peak predicted by Z2s is below the maximal CP point, δD < π/2 (or δD>−π/2) for the updated results. This corresponds to a positive cos δD with sa2>ca2 (θa > π/4) accordingly to Eq. (4) which is consistent with the preferred higher octant best fit of θa. The major peak moves above the maximal CP point for the Z‾2s case since the first parenthesis in Eq. (4) switches sign between the two residual symmetries. Moreover, the sum rule from the Z‾2s residual symmetry gives a broader δD distribution than the Z2s case.It is interesting to see that for all cases, a vanishing CP effect (δD is either 0 or π) is not highly disfavored by the two residual symmetries. Although the maximal CP phase point (δD=±π/2) does not have the highest probability since θa=π/4 is not favored by data, the two peaks are quite close to it. This is a very important and distinct feature that we can already conclude with the current data. The residual Z2s and Z‾2s symmetries prefer non-vanishing CP with probably a sizable value. Let us take perturbative expansion around the tribimaximal mixing pattern, θr ≡ δr is small and θa≡π/4+δa with both δr and δa denoting the deviations. Up to the linear order, Eq. (4) becomes [30],(7)Z2s:cosδD≈sscsδaδr,Z‾2s:cosδD≈−csssδaδr,The current best-fit gives δr=8.52∘ (8.57∘) and δa=3.50∘ (3.56∘) for NO (IO) [37]. For both mass orderings, the ratio between the two deviations δa/δr ≈ 0.4 cannot be compensated with the prefactor tan θs/2 ≈ 0.33 for Z2s or cotθs/2≈0.75 for Z‾2s to make cos δD close enough to 1. A non-vanishing leptonic Dirac CP phase is guaranteed by the residual Z2s and Z‾2s symmetries.Besides the theoretical prediction of δD distribution from the sum rule, the experimental side also provides a δD distribution. The first measurement of the Dirac CP phase was published by T2K [49] and NOvA [50] in 2019. Very recently, the T2K and NOvA collaborations have also published their combined analysis [33], shown as cyan dashed curves in Fig. 1, based on their recent measurements [51,52]. Correspondingly, the global fit group updated the NuFIT 6.0 result [37] in 2024 which is shown as purple dashed curves. The CP distributions are consistent between the global-fit result and T2K-NOvA joint analysis result in the case of IO, while for NO the CP distribution from the joint result is slightly shifted to the right-handed side. As shown in the figure, the best-fit values differ between the NO and IO cases. While IO prefers a maximal CP phase δD ≈ 280∘, NO prefers an almost vanishing one δD ≈ 180∘. Moreover, the NO has a broader distribution than the IO case due to the existing tension between T2K and NOvA results.To quantify the preference of the theoretical predictions of a model M by a CP measurement data set D, we take the Bayesian method [53] and define the following marginal likelihood,(8)P(D|M)≡∫P(D|fδM)P(sr2)P(ss2)P(sa2)dsr2dss2dsa2,where fδM≡δD(sr2,ss2,sa2)|M is the theoretical prediction from the residual symmetry as a function of the three mixing angles and P(D|fδM) is the probability distribution imposed by the data D. Comparing with Eq. (5), the δDp function for imposing a fixed-value for the Dirac CP phase which is a very special probability distribution is replaced by a more realistic P(D|fδM) with spread. If all the probability distribution functions P extracted from data follow the Gaussian distribution, after integration P(D|M) is actually a manifestation of the χmin2 that quantifies the extent of fitting a data set D with model M. A larger value of P(D|M), which corresponds to a smaller χmin2, means better fitting. To quantitatively compare the preference of data between the two residual symmetries, we adopt the Bayes factor, BF≡P(D|Z2s)/P(D|Z‾2s). To make it more explicit, a BF > 1 prefers Z2s and BF < 1 means that Z‾2s has a better change. The two models have equal preference when BF=1.Our result of the Bayesian factor is summarized in the BF (1D) column of Table 1

with inputs from the NuFIT 6.0 (T2K-NOvA joint analysis) [37] and the JUNO first data [32]. For NO, the global-fit result prefers Z‾2s which is consistent with Fig. 1 where the purple dashed curve has a larger overlap with the blue solid curve for Z‾2s than the red solid one for Z2s. In the case of IO, the data prefers Z2s since the purple dashed curve overlaps more with red solid curve than its blue solid counterpart. It is also interesting to see that the peak position of the red solid curve is consistent with the purple dashed one, which means the theoretically predicted δD value from Z2s is very consistent with the measurement. Note that the preference on the theoretical models with input from the T2K-NOvA joint analysis is slightly reduced than the global-fit case.

Moreover, since the global-fit result has a light preference for NO, we also present the combined Bayes factor,(9)P(D|M)≡pNO×P(D|M)NO+pIO×P(D|M)IO,to take into account the mass ordering weight factors pNO=0.574 and pIO=0.426.22

For the NuFIT 6.0 result, there is a Δχ2=0.6 preference for NO. This preference can be translated into priors pNO and pIO. Δχ2=χNO2−χIO2=0.6→LNO/LIO=exp(−0.3)=0.741. Fixing LNO=1, the relative thickness pNO=1/(1+0.741)=0.574 and pIO=0.741/(1+0.741)=0.426.

The combined mass ordering analysis shows that the global-fit result has only a mild preference for Z2s. The results in the table show opposite preferences between the NO and IO cases. We expect the mass ordering to be resolved with JUNO final result [15].4

Correlation between Dirac CP phase δD and atmospheric angle θa

Besides the predicted δD distribution from the sum rule as given in Eq. (6), it is interesting to consider the correlation behavior between δD and mixing angles. As reported in the current NuFIT result [37], the Dirac CP phase δD and the atmospheric angle sa2 are correlated, which can be seen in their two-dimensional χ2 distribution, shown as light (68% C.L.) and dark (95% C.L.) green contours in Fig. 2

. The best-fit points are shown with a red star for both mass orderings. There exists a slightly negative correlation between δD and sa2 for NO and a positive one for IO. Such information can also be useful for testing the residual Z2s and Z‾2s symmetries.

To obtain the correlation from the sum rules, we insert an additional δ(sa2−sa,fix2) to fix sa2 in Eq. (5). Then only uncertainties of sr2 and ss2 are taken into account. For each fixed sa2, one may obtain a 95% C.L. range for the sum rule prediction of δD. Varying the fixed sa2, a narrow band for the predicted δD as function of the atmospheric angle sa2 can be formed for Z2s (red) and Z‾2s (blue), respectively. One may see that the correlation behavior is opposite between Z2s and Z‾2s predictions which is independent on the mass ordering. The predicted δD deviates from π with an increasing sa2 for Z2s (red) and the opposite for Z‾2s (blue). Such difference in the correlation behavior arises from the different factors in Eq. (4), which is ss2−cs2sr2 for Z2s and ss2sr2−cs2 for Z‾2s. It becomes more explicit with the sign difference between the two expanded forms in Eq. (7). Moreover, the best-fit values and the corresponding uncertainties of sr2 and ss2 are almost the same between NO and IO. As a result, the blue or red contours in Fig. 2 are quite similar between the NO and IO cases. The octant of θa remains unknown and constitutes the dominant source of uncertainty in the prediction of δD. The next-generation experiments, such as DUNE [45] and T2HK [46], aim to resolve this octant degeneracy and significantly improve the precision of the δD prediction.When comparing the theoretical prediction with the global fit contours, it is interesting to see that the Zs2 symmetry has a positive correlation between δD and sa2 for the δD > π branch which is the consistent as the global-fit result for IO. In addition, the Z‾2s symmetry for δD > π has a positive correlation that is consistent with the global-fit result for NO.As mentioned above, we take the Bayesian factor to quantify the global-fit result preference on the theoretical models. Here we take the two-dimensional probability distribution of the atmospheric angle sa2 and the leptonic Dirac CP phase δD as data set D to evaluate P(D|M) in the similar way as defined in Eq. (8). In such a way, the correlation between sa2 and δD can be fully taken into account. The results are summarized in the BF (2D) column of Table 1. Taking this two-dimensional analysis into account, the preference of Z2s with NO decreases while the preference of Z‾2s increase in the IO case. This is consistent with the contour plots in Fig. 2 where Z‾2s (Z2s) has negative (positive) correlation for NO (IO) which are consistent with the NuFIT 6.0 contours. If both mass orderings are taken into a combined analysis, the preference for Z2s decreases.5

Conclusion

The residual symmetry is by definition the one that can survive symmetry breaking and apply to the neutrino mass matrix to directly dictate the mixing pattern. A unique prediction of the residual Z2s and Z‾2s symmetries is a correlation among the three mixing angles and the leptonic Dirac CP phase δD without involving model parameters. Later named as sum rule, such correlation involving only physical observables takes the same spirit as the correlation between weak gauge boson masses and the weak mixing angle that is predicted by the custodial symmetry that also survives the weak symmetry breaking.With the updated global fit and the JUNO first data release, the CP prediction by the custodial symmetries has much smaller uncertainty now. In particular, the vanishing CP case (δD=0 or π) is theoretically excluded now and the predicted CP distribution peaks around the maximal CP value (δD=±π2). Since the experimental measurements have dependence on the neutrino mass ordering, the residual Z2s and Z‾2s symmetries are preferred by IO and NO, respectively. The two-dimensional probability distribution of δD and the atmospheric angle θa with correlation can help distinguishing these two residual symmetries. Moreover, breaking the θa octant degeneracy remains crucial for giving more definite prediction of δD by eliminating the current double-peaked structure in the probability distributions to yield a single sharpe peak for each residual symmetry.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

The authors would like to thank Yue Meng for useful discussions. The authors are supported by the National Natural Science Foundation of China (12425506, 12375101, 12090060 and 12090064) and the SJTU First Class start-up fund (WF220442604). CFK is supported by IBS under the project code IBS-R018-D1. SFG is also an affiliate member of Kavli IPMU, University of Tokyo.

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