Resolving the octant of θ23 via radiative μ − τ symmetry breaking

Shu Luo1,* and Zhi-zhong Xing2,3,†1Department of Astronomy and Institute of Theoretical Physics and Astrophysics,

Xiamen University, Xiamen, Fujian 361005, China2Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, China

3Center for High Energy Physics, Peking University, Beijing 100080, China(Received 25 August 2014; revised manuscript received 19 September 2014; published 15 October 2014)

We point out that the observed neutrino mixing pattern at low energies is very likely to originate fromthe 3 × 3 Pontecorvo-Maki-Nakagawa-Sakata (PMNS) lepton flavor mixing matrix U which possessesthe exact μ-τ permutation symmetry jUμij ¼ jUτij (for i ¼ 1, 2, 3) at a superhigh energy scaleΛμτ ∼ 1014 GeV. The deviation of θ23 from 45° and that of δ from 270° in the standard parametrizationof U are therefore a natural consequence of small PMNS μ-τ symmetry breaking via the renormalization-group equations (RGEs) running from Λμτ down to the electroweak scale ΛEW ∼ 102 GeV. We find that theRGE-corrected value of θ23 is uniquely correlated with the neutrino mass ordering. In the minimalsupersymmetric standard model, the best-fit results θ23 ≃ 42.4° reported by Capozzi et al. (or θ23 ≃ 48.9°reported by Forero et al.) at ΛEW can arise from θ23 ¼ 45° at Λμτ if the neutrino mass ordering is inverted(or normal). Accordingly, the preliminary best-fit results of δ at ΛEW can also evolve from δ ¼ 270° at Λμτ

no matter whether the massive neutrinos are Dirac or Majorana particles.

DOI: 10.1103/PhysRevD.90.073005 PACS numbers: 14.60.Pq, 25.30.Pt

I. INTRODUCTION

From the discovery of atmospheric neutrino oscillationsin 1998 [1] until the observation of the smallest neutrinomixing angle θ13 in 2012 [2], experimental neutrinophysics was in full flourish. Today the era of precisionmeasurements has come. A number of undergoing andupcoming neutrino oscillation experiments aim to deter-mine the neutrino mass ordering, to probe the octant of thelargest neutrino mixing angle θ23, and to measure the DiracCP-violating phase δ. Such knowledge will be fundamen-tally important, as it can help identify the underlying flavorsymmetry or dynamics behind the observed pattern oflepton flavor mixing.As far as the octant of θ23 is concerned, it is desirable to

know whether this “atmospheric neutrino mixing” angledeviates from 45° or not, and if it does, how large or smallthe deviation is and in what direction the deviation evolves.A global analysis of current neutrino oscillation data doneby Capozzi et al. [3] yields the best-fit result θ23 ≃ 41.4°(normal neutrino mass ordering) or θ23 ≃ 42.4° (invertedneutrino mass ordering), which has a preference for the firstoctant (i.e., θ23 < 45°). In contrast, another best-fit resultreported by Forero et al. [4] is θ23 ≃ 48.8° (normal order-ing) or θ23 ≃ 49.2° (inverted ordering), by which thesecond octant (i.e., θ23 > 45°) is favored. In both casesθ23 ¼ 45° will be allowed when the 1σ or 2σ error bars aretaken into account. Hence the octant of θ23 remains an open

issue, and a resolution to this puzzle awaits more accurateexperimental data.On the other hand, the best-fit results of δ in both Ref. [3]

and Ref. [4] are close to an especially interesting value,270°, although the confidence level remains quite low. Infact, the T2K measurement of a relatively strong νμ → νeappearance signal [5] plays a crucial role in the global fit tomake θ13 consistent with the Daya Bay result [2] and drivea slight but intriguing preference for δ≃ 270° [3,4]. If thispreliminary expectation turns out to be true, it will beno problem to observe significant effects of leptonic CPviolation in the forthcoming long-baseline neutrino oscil-lation experiments.On the theoretical side, θ23 ¼ 45° and δ ¼ 270° are a

straightforward consequence of the μ-τ permutation sym-metry manifesting itself in the 3 × 3 Pontecorvo-Maki-Nakagawa-Sakata (PMNS) lepton flavor mixing matrixU [6], jUμij ¼ jUτij (for i ¼ 1, 2, 3), which can easily beembedded in an explicit flavor symmetry model. Hence thedeviation of θ23 from 45° and that of δ from 270° must berelated to small PMNS μ-τ symmetry breaking effects.This observation is important and suggestive, implying thatthe observed pattern of the PMNS matrix U should havean approximate μ-τ symmetry of the form jUμij≃ jUτij atlow energies [7]. In comparison, the Cabibbo-Kobayashi-Maskawa quark flavor mixing matrix V [8] does notpossess such a peculiar structure.In this paper we pay particular attention to a very real

possibility: the PMNS μ-τ symmetry is exact at a superhighenergy scale Λμτ where both tiny neutrino masses andlarge neutrino mixing angles could naturally be explainedin a well-founded theoretical framework (e.g., with the

*luoshu@xmu.edu.cn†xingzz@ihep.ac.cn

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canonical seesaw mechanism [9] and proper flavor sym-metry groups [10]). In this case we find that it is possible toresolve the octant of θ23 and the quadrant of δ via radiativeμ-τ symmetry breaking effects. Namely, the equalitiesjUμij ¼ jUτij are more or less violated when they evolvefrom Λμτ ∼ 1014 GeV down to the electroweak scaleΛEW ∼ 102 GeV via the relevant renormalization-groupequations (RGEs), such that the correct octant of θ23and the correct quadrant of δ can consequently be obtained.We carry out a numerical analysis of the issue for bothDirac and Majorana neutrinos based on the one-loop RGEsin the minimal supersymmetric standard model (MSSM).1

A striking finding of ours in fitting current neutrinooscillation data is that the RGE-corrected value of θ23is uniquely correlated with the neutrino mass ordering,

θ23 ≃ 42.4° reported in Ref. [3] (or θ23 ≃ 48.9° reported inRef. [4]) at ΛEW can evolve from θ23 ¼ 45° at Λμτ onlywhen the neutrino masses have an inverted (or normal)ordering. Accordingly, the preliminary best-fit results of δat ΛEW can also originate from δ ¼ 270° at Λμτ thanks toradiative μ-τ symmetry breaking. Such remarkable resultsare independent of any specific models of neutrino massgeneration and lepton flavor mixing, and they will soon betested in the upcoming precision experiments of neutrinooscillations.

II. THE RGEs of μ-τ SYMMETRY BREAKING

The PMNS lepton flavor mixing matrix can be para-metrized in the following “standard” way [12]:

U ¼

0B@

Ue1 Ue2 Ue3

Uμ1 Uμ2 Uμ3

Uτ1 Uτ2 Uτ3

1CA ¼

0B@

c12c13 s12c13 s13e−iδ

−s12c23 − c12s13s23eiδ c12c23 − s12s13s23eiδ c13s23s12s23 − c12s13c23eiδ −c12s23 − s12s13c23eiδ c13c23

1CAPν; ð1Þ

where cij ≡ cos θij, sij ≡ sin θij (for ij ¼ 12, 13, 23),δ is referred to as the Dirac CP-violating phase, andPν ¼ Diagfeiρ; eiσ; 1g contains two extra phase parametersif massive neutrinos are the Majorana particles. Up tonow θ12, θ13, and θ23 have all been measured to a good

degree of accuracy, and some preliminary hints for anontrivial value of δ have also been obtained from aglobal analysis of current neutrino oscillation data [3,4].Here we are concerned about the three PMNS μ-τ“asymmetries,”

Δ1 ≡ jUτ1j2 − jUμ1j2 ¼ ðcos2θ12sin2θ13 − sin2θ12Þ cos 2θ23 − sin 2θ12 sin θ13 sin 2θ23 cos δ;

Δ2 ≡ jUτ2j2 − jUμ2j2 ¼ ðsin2θ12sin2θ13 − cos2θ12Þ cos 2θ23 þ sin 2θ12 sin θ13 sin 2θ23 cos δ;

Δ3 ≡ jUτ3j2 − jUμ3j2 ¼ cos2θ13 cos 2θ23; ð2Þ

which satisfy the sum rule Δ1 þ Δ2 þ Δ3 ¼ 0. All thethree Δi vanish when the exact μ-τ permutation symmetryholds.We conjecture that the exact PMNS μ-τ symmetry

(i.e., Δi ¼ 0) can be realized at Λμτ ∼ 1014 GeV in a givenneutrino mass model with a proper flavor symmetry group[10]. In view of the facts that a nonzero and relatively largeθ13 has been observed and the preliminary best-fit value ofδ is not far from 270° at the electroweak scale [3,4], we

infer that the condition for all the three Δi to vanish shouldnaturally be θ23 ¼ 45° and δ ¼ 270° at the μ-τ symmetryscale.2 In this case Δi ≠ 0 can therefore be achieved atΛEW ∼ 102 GeV through the RGE running effects. Theone-loop RGEs of jUαij2 (for α ¼ e, μ, τ and i ¼ 1, 2, 3)have been derived by one of us in Ref. [14]. So it isstraightforward to write out the RGEs of Δi in the MSSMas follows.

1In this connection the standard-model RGEs are lessinteresting for two reasons: (a) it will be difficult to makethe deviation of θ23 from 45° appreciable even if the neutrinomasses are nearly degenerate and (b) the standard model itselfwill largely suffer from the vacuum-stability problem for themeasured value of the Higgs mass (≃125 GeV) as the energyscale is above 1010 GeV [11].

2Although Δi ¼ 0 might also result from θ23 ¼ 45° andθ13 ¼ 0° or θ23 ¼ 45° and δ ¼ 90° [13], neither of them is closeto the best-fit results of the lepton flavor mixing parametersreported in Refs. [3] and [4]. These two possibilities are muchless likely because they have to invoke violent RGE runningeffects between Λμτ and ΛEW in order to fit the presentexperimental data, which actually favor slight μ-τ symmetrybreaking [7]. That is why we concentrate our interest only on thepossibility of θ23 ¼ 45° and δ ¼ 270° in this paper.

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A. Dirac neutrinos

If massive neutrinos are the Dirac particles, we find

16π2dΔ1

dt¼ −y2τ ½ξ21ðjUτ1j2Δ2 þ jUτ2j2Δ1 þ jUe3j2Þ þ ξ31ðjUτ1j2Δ3 þ jUτ3j2Δ1 þ jUe2j2Þ�;

16π2dΔ2

dt¼ þy2τ ½ξ21ðjUτ1j2Δ2 þ jUτ2j2Δ1 þ jUe3j2Þ − ξ32ðjUτ2j2Δ3 þ jUτ3j2Δ2 þ jUe1j2Þ�;

16π2dΔ3

dt¼ þy2τ ½ξ31ðjUτ1j2Δ3 þ jUτ3j2Δ1 þ jUe2j2Þ þ ξ32ðjUτ2j2Δ3 þ jUτ3j2Δ2 þ jUe1j2Þ�; ð3Þ

where t≡ lnðμ=ΛμτÞwith μ being an arbitrary scale between ΛEW and Λμτ, y2τ ¼ ð1þ tan2 βÞm2τ=v2 is the Yukawa coupling

eigenvalue of the tau lepton in the MSSM with tan β and v being self-explaining, and ξij ≡ ðm2i þm2

jÞ=Δm2ij, with

Δm2ij ≡m2

i −m2j being the neutrino mass-squared differences. Given the fact jΔm2

31j≃ jΔm232j ∼ 30Δm2

21 withΔm2

21 ≃ 7.5 × 10−5 eV2, ξ21 ≫ jξ31j≃ jξ32j is expected to hold in most cases. But this does not necessarily mean thatthe μ-τ asymmetry Δ3 should be more stable against radiative corrections than the other two asymmetries. The reason issimply that the running behaviors of Δi depend also on the initial inputs of all the nine jUαij2. In general, however, anappreciable deviation of θ23 from 45° (i.e., an appreciable deviation of Δ3 from zero) requires a sufficiently large value oftan β, and its evolving direction is governed by the neutrino mass ordering or equivalently the sign of Δm2

31 or Δm232.

B. Majorana neutrinos

If massive neutrinos are the Majorana particles, we arrive at

16π2dΔ1

dt¼ −y2τfξ21ðjUτ1j2Δ2 þ jUτ2j2Δ1 þ jUe3j2Þ þ ξ31ðjUτ1j2Δ3 þ jUτ3j2Δ1 þ jUe2j2Þþ ζ21½ðjUτ1j2Δ2 þ jUτ2j2Δ1 þ jUe3j2Þ cosΦ12 þ J sinΦ12�þ ζ31½ðjUτ1j2Δ3 þ jUτ3j2Δ1 þ jUe2j2Þ cosΦ13 − J sinΦ13�g;

16π2dΔ2

dt¼ þy2τfξ21ðjUτ1j2Δ2 þ jUτ2j2Δ1 þ jUe3j2Þ − ξ32ðjUτ2j2Δ3 þ jUτ3j2Δ2 þ jUe1j2Þþ ζ21½ðjUτ1j2Δ2 þ jUτ2j2Δ1 þ jUe3j2Þ cosΦ12 þ J sinΦ12�− ζ32½ðjUτ2j2Δ3 þ jUτ3j2Δ2 þ jUe1j2Þ cosΦ23 þ J sinΦ23�g;

16π2dΔ3

dt¼ þy2τfξ31ðjUτ1j2Δ3 þ jUτ3j2Δ1 þ jUe2j2Þ þ ξ32ðjUτ2j2Δ3 þ jUτ3j2Δ2 þ jUe1j2Þþ ζ31½ðjUτ1j2Δ3 þ jUτ3j2Δ1 þ jUe2j2Þ cosΦ13 − J sinΦ13�þ ζ32½ðjUτ2j2Δ3 þ jUτ3j2Δ2 þ jUe1j2Þ cosΦ23 þ J sinΦ23�g; ð4Þ

where ζij≡2mimj=Δm2ij, cosΦij≡ReðUτiU�

τjÞ2=jUτiU�τjj2,

sinΦij ≡ ImðUτiU�τjÞ2=jUτiU�

τjj2, and the leptonic Jarlskoginvariant J [15] is defined through

ImðUαiUβjU�αjU

�βiÞ ¼ J

Xγ

ϵαβγXk

ϵijk ð5Þ

with the greek and latin subscripts running over (e, μ, τ) and(1,2,3), respectively. Since the sign of ζij is always thesame as that of ξij, it is possible to adjust the evolvingdirection of Δ3 without much fine-tuning of the otherrelevant parameters. Hence, similar to the Dirac neutrino

case, the RGE-triggered deviation of θ23 from 45∘ might beclosely correlated with the neutrino mass ordering in theMajorana case.In Eq. (4) it should be noted that the two Majorana

CP-violating phases ρ and σ in the standard parametriza-tion of U affect the running behaviors of Δi via cosΦijand sinΦij. One should also note that J ¼ðsin 2θ12 sin 2θ13 cos θ13 sin 2θ23 sin δÞ=8, which onlydepends on the Dirac CP-violating phase δ, measuresthe strength of CP violation in neutrino oscillations.Therefore, δ ∼ 270° is especially favorable for significantCP-violating effects in the lepton sector, no matter whetherthe massive neutrinos are Dirac or Majorana particles.

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III. NUMERICAL RESULTS FOR Δi, θ23, and δ

We proceed to numerically illustrate the effects ofμ-τ symmetry breaking regarding the PMNS matrix U—namely, the quantities Δi run from Δi ¼ 0 (i.e., δ23 ¼ 45°and δ ¼ 270°) at Λμτ ∼ 1014 GeV down to ΛEW ∼ 102 GeVvia the one-loop RGEs obtained in Eq. (3) or (4). Given aproper value of tan β, the values of m1, Δm2

21, Δm231, θ12,

and θ13 at Λμτ should be carefully chosen such that the best-fit results of Δm2

21, Δm231, θ12, θ13, θ23, and δ at ΛEW as

listed in Table I can all be reproduced to a good degree ofaccuracy.3 If this strategy is workable, then the deviation ofθ23 from 45° and that of δ from 270° will be purelyattributed to the RGE-triggered PMNS μ-τ symmetrybreaking effects.4

A. Dirac neutrinos

For simplicity, we fix tan β ¼ 31 and input m1 ¼ 0.1 eVat Λμτ, where Δ1, Δ2 and Δ3 are vanishing (or equivalently,θ23 ¼ 45° and δ ¼ 90°), in our numerical calculations.Table II shows the input and output values of all therelevant parameters for two examples, which are based onthe best-fit results reported by Capozzi et al. [3] and Foreroet al. [4], respectively. Figure 1 illustrates how Δi evolvein either example. Some comments and discussions arein order.(1) Given the inverted neutrino mass ordering, the best-

fit results of the six neutrino oscillation parameters Δm221,

Δm231, θ12, θ13, θ23 and δ at ΛEW in Example I [3] can

successfully be reproduced from the proper inputs atΛμτ. Inthis case θ23ðΛEWÞ lies in the first octant, and θ23ðΛμτÞ −θ23ðΛEWÞ≃ 2.6° holds thanks to the RGE running effect.At the same time, we obtain δðΛμτÞ − δðΛEWÞ≃ 34°.Hence the RGE evolution can also provide a resolutionto the quadrant of δ.(2) In contrast, only the normal neutrino mass ordering

allows us to obtain θ23ðΛEWÞ≃ 48.4° from θ23ðΛμτÞ ¼ 45°via the RGE evolution as shown in Example II [4].Moreover, we obtain δðΛEWÞ≃237° from δðΛμτÞ ¼ 270°,and this result is also consistent with the correspondingbest-fit value δ≃ 241° as listed in Table I. The futureexperimental data will only verify one of the above two

TABLE II. The RGE-triggered PMNS μ-τ symmetry breaking effects for Dirac neutrinos running from Δi ¼ 0 atΛμτ ∼ 1014 GeV down to ΛEW ∼ 102 GeV in the MSSM with tan β ¼ 31.

Example I (Capozzi et al. [3]) Example II (Forero et al. [4])

Parameter Input (Λμτ) Output (ΛEW) Input (Λμτ) Output (ΛEW)

m1 (eV) 0.100 0.093 0.100 0.093Δm2

21 (eV2) 1.82 × 10−4 7.54 × 10−5 1.96 × 10−4 7.60 × 10−5

Δm231 (eV2) −2.60 × 10−3 −2.34 × 10−3 3.00 × 10−3 2.48 × 10−3

θ12 10.8° 33.6° 10.3° 34.6°θ13 9.4° 8.9° 8.4° 8.8°θ23 45.0° 42.4° 45.0° 48.4°δ 270° 236° 270° 237°J −0.015 −0.029 −0.013 −0.029Δ1 0 0.053 0 0.114Δ2 0 −0.141 0 0.001Δ3 0 0.088 0 −0.115

TABLE I. The best-fit values of Δm221, Δm2

31, θ12, θ13, θ23 and δ obtained from two recent global analyses of current neutrinooscillation data [3,4].

Reference Mass ordering Δm221 (eV2) Δm2

31 (eV2) θ12 θ13 θ23 δ

Capozzi et al. [3] Normal7.54 × 10−5

þ2.47 × 10−3 33.7° 8.8° 41.4° 250°Inverted −2.34 × 10−3 8.9° 42.4° 236°

Forero et al. [4] Normal7.60 × 10−5

þ2.48 × 10−3 34.6° 8.8° 48.9° 241°Inverted −2.38 × 10−3 8.9° 49.2° 266°

3Note that the notations δm2 ≡m22 −m2

1 and Δm2 ≡m23 −ðm2

1 þm22Þ=2 have been used in Ref. [3]. They are related with

Δm221 and Δm2

31 as follows: Δm221 ¼ δm2 and Δm2

31 ¼Δm2 þ δm2=2.

4Note that the νfit group’s best-fit results [16] are not taken intoaccount in our numerical examples, because they happen tocorrespond to the disfavored cases listed in Table I (i.e., θ23 < 45°for the normal neutrino mass ordering or θ23 > 45° for theinverted ordering, in conflict with our expectations shown inTables II and III, respectively).

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possibilities for the octant of θ23, but it will be interestingto test the expected correlation between the neutrinomass ordering and the deviation of θ23 (or δ) from 45°(or 270°).(3) Figure 1 shows the behaviors of three PMNS

μ-τ asymmetries Δi evolving from Λμτ down to ΛEWfor the two examples under discussion. In view of Δ3 ¼cos2 θ13 cos 2θ23 in Eq. (2), one must have Δ3ðΛEWÞ > 0for θ23ðΛEWÞ < 45° in Example I, and Δ3ðΛEWÞ < 0 forθ23ðΛEWÞ > 45° in Example II. In comparison, the run-ning behaviors of Δ1 and Δ2 are not so straightforward,because they depend on all the three flavor mixing anglesand the CP-violating phase δ. But Δ1 þ Δ2 þ Δ3 ¼ 0holds at any energy scale between ΛEW and Λμτ, as one cansee in Fig. 1.

B. Majorana neutrinos

In this case we simply fix tan β ¼ 30 and input m1 ¼0.1 eV at Λμτ, where Δ1 ¼ Δ2 ¼ Δ3 ¼ 0 holds, in ournumerical calculations. Table III is a brief summary of theinput and output values of all the relevant parameters forExample I [3] and Example II [4], respectively. In addition,Fig. 2 illustrates how the three PMNS μ-τ asymmetriesevolve from Λμτ down to ΛEW in either example.Although the present case involves two extra CP-violating

phases ρ and σ, the running behaviors of Δi in Fig. 2 arequite similar to those in Fig. 1. Of course, one has to adjustthe initial values of ρ and σ at Λμτ in a careful way, such thatthe best-fit results of the six neutrino oscillation parameterscan correctly be reproduced at ΛEW. We find that it is reallypossible to resolve the octant of θ23 and the quadrant of δ atthe same time via radiative PMNS μ-τ symmetry breaking.Very similar to the Dirac neutrino case, the RGE-triggereddeviation of θ23 from 45° in the Majorana case is alsoclosely correlated with the neutrino mass ordering. Namely,θ23 ≃ 42.4° reported in Ref. [3] (or θ23 ≃ 48.9° reported inRef. [4]) at ΛEW can evolve from θ23 ¼ 45∘ at Λμτ onlywhen the neutrino masses have an inverted (or normal)ordering.In view of the fact that the present best-fit results

of θ23 and δ are still quite preliminary, we foresee thatthey must undergo some changes before they are welldetermined by the more precise experimental data in thenear future. Hence our numerical analysis is not targetedfor a complete parameter-space exploration but mainly forthe purpose of illustration [17]. Its outcome supports our

TABLE III. The RGE-triggered PMNS μ-τ symmetry breakingeffects for Majorana neutrinos running from Δi ¼ 0 at Λμτ ∼1014 GeV down to ΛEW ∼ 102 GeV in the MSSM withtan β ¼ 30.

Example I(Capozzi et al. [3])

Example II(Forero et al. [4])

Parameter Input(Λμτ)

Output(ΛEW)

Input(Λμτ)

Output(ΛEW)

m1 (eV) 0.100 0.087 0.100 0.087Δm2

21

(eV2)1.70×10−4 7.54×10−5 2.12×10−4 7.60×10−5

Δm231

(eV2)−2.98×10−3 −2.34×10−3 3.50×10−3 2.48×10−3

θ12 35.2° 33.7° 32.1° 34.6°θ13 10.1° 8.9° 6.9° 8.8°θ23 45.0° 42.4° 45.0° 48.9°δ 270° 236° 270° 241°ρ −82° −66° −76° −45°σ 19° 27° 17° 29°J −0.040 −0.029 −0.027 −0.030Δ1 0 0.054 0 0.111Δ2 0 −0.142 0 0.022Δ3 0 0.088 0 −0.133

2 4 6 8 10 12 14

−0.12

−0.08

−0.04

0

0.04

0.08

0.12

log μ (GeV)

Example IΔ

3

Δ1

Δ2

2 4 6 8 10 12 14

−0.12

−0.08

−0.04

0

0.04

0.08

0.12

log μ (GeV)

Example II

Δ2

Δ3

Δ1

FIG. 1 (color online). TheRGE-triggeredμ-τ symmetry breakingeffects for Dirac neutrinos running fromΔi ¼ 0 atΛμτ ∼ 1014 GeVdown to ΛEW ∼ 102 GeV in the MSSM with tan β ¼ 31.

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original conjecture: the slight μ-τ symmetry breakingbehind the observed pattern of lepton flavor mixing canoriginate from the RGE evolution from a superhigh flavorsymmetry scale down to the electroweak scale. Note thatthere are two adjustable unknown parameters in ourcalculations: the absolute neutrino mass m1 and theMSSM parameter tan β. Once m1 is experimentally deter-mined and tan β is theoretically fixed, for example, it willbe interesting to see whether one can still resolve the octantof θ23 and the quadrant of δ with the help of radiativePMNS μ-τ symmetry breaking effects.We admit that the present best-fit result δ ∼ 270° remains

too preliminary. In fact, there is not any nontrivial regionassociated with the allowed values of δ at the 2σ level [3,4].Hence it also makes sense to look at the RGE-triggered

corrections to θ23 ¼ 45° and δ ¼ 90° for the energy scale toevolve from Λμτ down to ΛEW. This possibility has alreadybeen discussed in some literature (see, e.g., Refs. [17,18]).Once the CP-violating phase δ is measured or constrainedto a better degree of accuracy in the near future, it will bepossible to examine whether the quantum corrections canreally accommodate the observed effect of PMNS μ-τsymmetry breaking or not.

IV. SUMMARY AND FURTHER DISCUSSIONS

To summarize, we have conjectured that the PMNS μ-τpermutation symmetry is exact at a superhigh energy scaleΛμτ ∼ 1014 GeV, where the origin of neutrino masses andflavor mixing has a good dynamic reason, and its slightbreaking happens via the RGE running down to theelectroweak scale ΛEW ∼ 102 GeV. This idea is particularlyinteresting in the sense that it can help resolve the octantof θ23 and the quadrant of δ at the same time thanks toradiative PMNS μ-τ symmetry breaking in the MSSM. Infitting current neutrino oscillation data we have found thatthe RGE-triggered deviation of θ23 from 45° is uniquelycorrelated with the neutrino mass ordering, θ23 ≃ 42.4° [3](or θ23 ≃ 48.9° [4]) at ΛEW can naturally originate fromθ23 ¼ 45° at Λμτ if the neutrino mass ordering is inverted(or normal). Accordingly, the preliminary best-fit resultsof δ at ΛEW can also evolve from δ ¼ 270° at Λμτ. Suchremarkable findings are independent of any specific modelsof neutrino mass generation and lepton flavor mixing, andthey will soon be tested in the upcoming neutrino oscil-lation experiments.Note that some previous studies of the RGE evolution of

lepton flavor mixing parameters have more or less involvedthe μ-τ symmetry breaking effects [19]. In this connection afew constant neutrino mixing patterns which possessjUμij ¼ jUτij, such as the bimaximal [20] and tribimaximal[21] ones with θ13 ¼ 0° and θ23 ¼ 45°, have been assumedat a superhigh energy scale, and their RGE runningbehaviors have been investigated mainly to see whethera finite θ13 can be radiatively generated at low energies[18]. The closest example of this kind should be the work[22] on radiative corrections to the tetramaximal neutrinomixing pattern [23], in which θ13 ≃ 8.4°, θ23 ¼ 45° andδ ¼ 90° or 270° have been predicted. Our present work isdifferent from the previous ones in several aspects: (a) it isnot subject to any explicit neutrino mixing pattern, (b) itfocuses on the PMNS μ-τ asymmetries Δi and its RGEevolution, (c) it provides a reasonable resolution to theoctant of θ23 by attributing it to the PMNS μ-τ symmetrybreaking effect, and (d) it may also resolve the quadrant of δin a similar way. All in all, we have established the RGEconnection between a given neutrino mass model with theexact μ-τ symmetry at superhigh energies and the neutrinooscillation parameters at low energies. Such a connection isexpected to be very useful for neutrino phenomenology inthe era of precision measurements.

2 4 6 8 10 12 14

−0.12

−0.08

−0.04

0

0.04

0.08

0.12

log μ (GeV)

Example IΔ

3

Δ2

Δ1

2 4 6 8 10 12 14

−0.12

−0.08

−0.04

0

0.04

0.08

0.12

log μ (GeV)

Example IIΔ

1

Δ2

Δ3

FIG. 2 (color online). The RGE-triggered μ-τ symmetry break-ing effects for Majorana neutrinos running from Δi ¼ 0 at Λμτ ∼1014 GeV down toΛEW ∼ 102 GeV in theMSSMwith tan β ¼ 30.

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ACKNOWLEDGMENTS

This work was supported in part by the National Natural Science Foundation of China under Grants No. 11105113 (S. L.)and No. 11135009 (Z. Z. X.).

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