Determination of the mixing between active neutrinos and sterile neutrinothrough the quark-lepton complementarity and self-complementarity

Hong-Wei Ke,1,* Tan Liu,1 and Xue-Qian Li2,†1School of Science, Tianjin University, Tianjin 300072, China2School of Physics, Nankai University, Tianjin 300071, China(Received 11 August 2014; published 19 September 2014)

It is suggested that there is an underlying symmetry which relates the quark and lepton sectors. Namely,among the mixing matrix elements of Cabibbo-Kobayashi-Maskawa for quarks and Pontecorvo-Maki-Nakawaga-Sakata for leptons there exist complementarity relations at a high energy scale (such as theseesaw or even the grand unification theory scales). We assume that the relations would remain during thematrix elements running down to the electroweak scale. Observable breaking of the rational relation isattributed to the existence of sterile neutrinos that mix with the active neutrino to result in the observablePontecorvo-Maki-Nakawaga-Sakata matrix. We show that involvement of a sterile in the (3þ 1)model induces that jUe4j2 ¼ 0.040, jUμ4j2 ¼ 0.009, and sin22α ¼ 0.067. We also find a new self-complementarity ϑ12 þ ϑ23 þ ϑ13 þ α ≈ 90°. The numbers are generally consistent with those obtained byfitting recent measurements, especially in this scenario, to the existence of a sterile neutrino that does notupset the LEP data; i.e., the number of neutrino types is very close to 3.

DOI: 10.1103/PhysRevD.90.053009 14.60.Pq, 12.15.Ff, 14.60.Lm, 14.60.St

I. INTRODUCTION

By the recent observation, the neutrino masses are muchlighter than the corresponding leptons, but the origin of theneutrino masses remains a mystery. Even so, in analogto the quark sector, the neutrino flavor eigenstates aredifferent from their mass eigenstates, so the Pontecorvo-Maki-Nakawaga-Sakata (PMNS) matrix appears [1,2]. Onthe aspect, the quark sector has been treated separately, anda corresponding mixing matrix Cabibbo-Kobayashi-Maskawa (CKM) [3–5] plays the role. It is well knownthat to cancel the gauge anomaly, quark and lepton sectorsmust exist simultaneously and have the same generations.Therefore, one is tempted by the one-to-one correspon-dence of quark and lepton sectors to conjecture there mightbe some intimate relations between the two sectors. Eventhough at the practical world, the two sectors look quitedifferent, one may consider that at very high scales such asthe seesaw, grand unification, or even Planck scales, theyoriginate from the same or at least related sources.Therefore there might exist a large symmetry that relatesthe two sectors. Indeed, at the high energy scale of about1015 GeV, where the strong, electromagnetic, and weakinteractions are unified into a large symmetry such asSU(5), SO(10), E6, etc. [6], quarks and leptons may residein the same representations of the large group. Therefore bythe grand unification theories it is natural to expect thatsuch complementary relations may exist. Earlier in 2004,Raidal [7] suggested that grand unification theories mayrelate the quark and lepton sectors and predicted some

phenomenological consequences, and Ma et al. [8] con-sidered it as the theoretical base of complementarity.If the two sectors indeed originate from a large sym-

metry, even though during the process of running downfrom a higher energy scale to our practical electroweakscale many quantities look different, some of the relationsmay remain.With a certain parametrization the quark-lepton

complementarity [9] and self-complementarity [10–13]are noticed. Indeed such relations are approximate.Motivated by the picture described above, we assume thatthe complementarity and self-complementarity are exactand guaranteed by the residual symmetry even though it isnot clear yet. On other aspect, the experimental measure-ments show that such relations are only approximate. Onemay think that such deviations are due to measurementerrors, or there exists new physics whose existence resultsin the declination from exact complementarity and self-complementarity. The goal of this work is to search for apossible new physics scenario that may cause such adeclination.The short-baseline neutrino oscillations indicate there

may exist light sterile neutrinos if CPT invariance isconserved [14–16]. The sterile neutrinos do not directlyparticipate in weak interaction, but may mix with the activeneutrinos of three generations. Therefore they would makesubstantial contributions to the observable physical quan-tities via the mixing.The mixing among active neutrinos and sterile neutrinos

produces an extended 4 × 4 PMNS matrix [17,18] withcertain parameters. In the previous 3 × 3 PMNS matrix[8,11,12] the mixing with sterile neutrinos was notincluded, so that the quark-lepton complementarity and

*khw020056@hotmail.com†lixq@nankai.edu.cn

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self-complementarity are approximate. Now we assume thequark-lepton complementarity and self-complementarityto be exact, whereas the mixing among active and sterileneutrinos causes the apparent declination. The startingpoint of this work: the mixing angles for quarks andleptons possess an exact complementarity, but are con-taminated by the existence of sterile neutrinos.In this paper we will employ the scenario with three

active neutrinos plus one sterile neutrino (3þ 1). Thus wemay fix mixing angles between the light sterile neutrinowith the active ones in terms of the assumed quark-leptoncomplementarity and self-complementarity. Though thescenario is simple, its prediction generally coincides withthe experimental observation; thus the present data do notsuggest to us to abandon the simple version [19].To be explicit, we restate our strategy as follows: By

supposing the quark-lepton complementarity and self-complementarity to be exact, we calculate the mixingmatrix elements jUe1j, jUe2j, jUe3j, jUμ3j, and jUτ3j ofthe original matrix PMNS (i.e., without mixing with thesterile neutrino). Then the mixing matrix is extended to a4 × 4 matrix that includes mixing between active neutrinosand a sterile neutrino as suggested in literature, and the3 × 3 submatrix at the left-upper corner of the 4 × 4 matrixis the practical PMNS matrix. Comparing the matrixelements with the data one can fix the mixing anglesbetween the sterile neutrino and the active neutrinos, anddetermine the weak CP phase (see the later context fordetails).Finally, we test the scenario by calculating the number of

neutrino generations, which is determined by the LEP datavery accurately as very close to 3. Our result shows that thisnumber is perfectly respected in the new scenario.The paper is organized as follows. After the Introduction

we describe our detailed strategy and derivation of relevant

formulas in Sec. II. In Sec. III, we present our numericalresults along with all the inputs and discuss both exper-imental and theoretical errors. In Sec. IV we will make asummary.

II. THE MIXING OF FERMIONS ANDQUARK-LEPTON COMPLEMENTARITY

AND SELF-COMPLEMENTARITY

In this section we show explicitly how to fix the mixingangles between the active neutrinos and sterile neutrino andthe CP phase under the hypothesis of the exact quark-lepton complementarity and self-complementarity.

A. The mixing of fermions in SM

The mixing among quarks or leptons is described by theCKM and PMNS matrices that appear in the weak chargedcurrents. The quark sector involves the u-type and d-typequarks, whereas the leptonic sector involves neutrinos andcharged leptons. The relevant Lagrangian is

L¼ gffiffiffi2

p ULγμVCKMDLWþ

μ −gffiffiffi2

p ELγμVPMNSNLWþ

μ þH:c:;

ð1Þ

where UL ¼ ðuL; cL; tLÞT , DL ¼ ðdL; sL; bLÞT , EL ¼ðeL; μL; τLÞT , and NL ¼ ðν1; ν2; ν3ÞT . VCKM and VPMNSare the CKM and PMNS matrices, respectively. If therewere no sterile neutrino, both quark and lepton sectorscontain three generations, so their mixing matrices aresimilar. As it is well known that real physics is independentof any parametrization schemes, so it is convenient to setVCMS and VPMNS in the P1 parametrization [8] as

V ¼

0B@

Ue1 Ue2 Ue3

Uμ1 Uμ2 Uμ3

Uν1 Uν2 Uν3

1CA ¼

0B@

c12c13 s12s13 s13−c12s23s13 − s12c23eiδðδ

0Þ −s12s23s13 þ c12c23eiδðδ0Þ s23c13

−c12s23s13 þ s12s23eiδðδ0Þ −s12s23s13 − c12s23eiδðδ

0Þ c23c13

1CA: ð2Þ

Here sij and cij denote sin θijðsinϑijÞ and cos θijðcos ϑijÞwith i, j ¼ 1, 2, 3. In this work, we use θij for the quarksector and ϑij for the lepton sector, respectively.Thanks to hard experimental measurements on the weak

processes where the CKM matrix is involved, the mixingparameters for the quark sector are more precisely fixed andtheir central values [12] are

θ12 ¼ 13.023°; θ23 ¼ 2.360°;

θ13 ¼ 0.201°; δ ¼ 69.10°: ð3ÞDefinitely, certain experimental errors still exist, andthey would cause theoretical uncertainties in our

predictions on the PMNS parameters. We will discuss thatissue later.The parameters in the 3 × 3 PMNS matrix that are

determined by the measured data [12] are

ϑ12 ¼ 33.65°; ϑ23 ¼ 38.41°; ϑ13 ¼ 8.93°; ð4Þ

which are directly measured by the neutrino-involvedexperiments, especially the neutrino oscillations.As we discussed above, among the CKM and PMNS

matrix elements, there are complementarity and self-complementarity relations. In the P1 parametrization,the relations reduce to some direct relations among the

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mixing angles. The quark-lepton complementarity suggestsθ12þϑ12≈45°, θ23þϑ23≈45° and the self-complementarityrequires ϑ12 þ ϑ13 ≈ ϑ23 to be held.Comparing with data, one immediately notices that

even though those relations are in a good approximation,obvious deviation of the obtained mixing matrix from thedata,

V 0 ¼

0B@

U0e1 U0

e2 U0e3

U0μ1 U0

μ2 U0μ3

U0ν1 U0

ν2 U0ν3

1CA; ð5Þ

demands an explanation. That deviation happens in thescenario with only three types of neutrinos as required bythe standard model, so when the theory is extended toinvolve new components, the problem will easily be solved.

B. The mixing of neutrinos beyond SM

In some previous works, the authors introduced one ormore sterile neutrinos to explain the data of short-baselineneutrino oscillation [14–16]. In this work we consider themodel of three active neutrinos mixing with one sterileneutrino (νs). The sterile neutrino does not directly par-ticipate in the weak interaction, so before taking intoaccount its mixing with active neutrinos, the weak inter-action Lagrangian for the leptonic sector is

−gffiffiffi2

p ð eL μL τL Þγμ0B@

U0e1 U0

e2 U0e3 0

U0μ1 U0

μ2 U0μ3 0

U0τ1 U0

τ2 U0τ3 0

1CA

×

0BBB@

ν1

ν2

ν3

νs

1CCCA

L

Wþμ þ H:c: ð6Þ

Apparently as the active neutrinos mix with the sterileneutrino, the values of the mixing matrix elements inEqs. (5) and (6) are definitely affected. Once appropriatemixing parameters are chosen, these modified mixingmatrix elements may coincide with the available data.By contrast, if one cannot fix a set of such mixingparameters to make the new matrix elements to meet thedata, the model would fail. Later we will show that theadopted scenario succeeds; i.e., the newly obtained PMNSmatrix elements are generally consistent with the data, andthe theoretical uncertainties are smaller than the experi-mental errors.To account for the possible mixing between the sterile

neutrino and the active ones, we introduce a 4 × 4 matrix.In a complete picture, the mixing of neutrinos (3 activeneutrinosþ 1 sterile neutrino) could be

0BBB@

νe

νμ

ντ

νs

1CCCA ¼ V4×4

0BBB@

ν1

ν2

ν3

ν4

1CCCA; ð7Þ

where νs is the sterile neutrino and the extended 4 × 4matrix is written as

V4×4 ¼

0BBB@

U00e1 U00

e2 U00e3 U00

e4

U00μ1 U00

μ2 U00μ3 U00

μ4

U00τ1 U00

τ2 U00τ3 U00

τ4

U00s1 U00

s2 U00s3 U00

s4

1CCCA; ð8Þ

which can be realized in a rotation [17,18]

V4×4 ¼ R23ϕR13R12R14R24R34; ð9Þ

and the relevant matrices R23; R13; R12; R14; R24; R34, and ϕare simple and straightforward as done in the literature;however, for the readers’ convenience we present them inthe Appendix. It is noted that the left upper 3 × 3 submatrixcorresponds to the measured PMNS mixing matrix whoseelements are fixed by the neutrino oscillation experiments.

C. The strategy to fix the mixing parameters

As a sterile neutrino is introduced into the model, therewould be more free parameters. We have obtained themodules of U0

e1, U0e2, U

0e3, U

0μ3, and U0

ν3 in Eq. (5). Takinginto account the mixing with the sterile neutrino, theelements U0

e1, U0e2, U

0e3, U

0μ3, and U0

ν3 are modified toU00

e1, U00e2, U

00e3, U

00μ3, and U00

ν3 in Eq. (8). By adjusting themixing parameter, we can make those elements eventuallycoincide with the measured values.Supposing that the quark-lepton complementarity

and self-complementarity hold for the three-generationneutrino structure, the central values of the measuredCKM matrix elements for quarks would fully determineθ12 ¼ ð13.023� 0.038Þ°, θ23¼ð2.360�0.052Þ°, andθ13 ¼ ð0.201� 0.009Þ°; then we can obtain ϑ012 ¼ð31.977� 0.038Þ°, ϑ023 ¼ ð42.640� 0.052Þ°, ϑ013 ¼ð10.663� 0.014Þ°, which are deviated from the valuesgiven in Eq. (4). Let us rewrite the PMNSmatrix in terms ofthe obtained angles as

jV 0j ¼

0B@0.834� 0.001 0.520� 0.001 0.185� 0.001

− − 0.666� 0.001

− − 0.723� 0.001

1CA:

ð10Þ

The corresponding experimental values in the 3 × 3 VPMNSis [12]

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jVPMNSj¼

0B@0.822�0.011 0.547�0.016 0.155�0.008

− − 0.614�0.018

− − 0.774�0.014

1CA:

ð11Þ

One can notice the deviation.Then we introduce the mixing with the sterile neutrino

and recalculate the modules of U00e1, U

00e2, U

00e3, U

00μ3, and U

00τ3

in the V4×4 matrix. Now the numbers can be compared withthe measured values of the VPMNS elements. Here let usexplicitly show the expression of jUe1j as an example

jU00e1j ¼ cosϑ012 cos ϑ

013 cos α ¼ 0.834 cos α: ð12Þ

Comparing with the data,

jU00e1j ¼ jUe1j; ð13Þ

we fix the mixing parameters. The other elements and CPphase δ0 are simultaneously fixed, when the χ2 methods areemployed [20,21].At last, using these parameters we complete the gener-

alized and practical 4 × 4 matrix, and its left-upper 3 × 3submatrix is just the practical matrix jVPMNSj.

III. NUMERICAL RESULTS

There are two possible schemes for the 3þ 1 mixing.(1) The first scheme: The sterile neutrino mixes with the

three active neutrino by different mixing parameters,namely there are three free parameters α, β, and γ.To fit the data, we set the values as

α ¼ ð0.00� 0.02Þ°, β ¼ ð14.19� 0.18Þ°, γ ¼ð12.46� 0.19Þ°, and CP phase δ0 ¼ð0.00� 0.01Þ°. The module of the PMNS matrixreads

jV4×4j ¼

0BBB@

0.834� 0.001 0.505� 0.001 0.153� 0.002 0.165� 0.002

0.496� 0.001 0.541� 0.001 0.621� 0.002 0.277� 0.003

0.243� 0.001 0.627� 0.001 0.740� 0.001 0.001� 0.004

0� 0.001 0.245� 0.004 0.209� 0.004 0.947� 0.002

1CCCA: ð14Þ

The resultant jU00e3j, jU00

μ3j, and jU00τ3j are close to the

data. Based on our calculations we have jU00e4j2 ¼

0.027� 0.004, jU00μ4j2 ¼ 0.077� 0.006, and sin22α ¼

0� 0.002. In the earlier works [14,19,22,23] the authorscarried out an analysis of short-baseline neutrino oscilla-tions in the 3þ 1 neutrino mixing scenario. Their resultsare presented in Table I.

(2) The second scheme: This is a simplified version ofthe first scheme, we let α ¼ β ¼ γ as discussed inRef. [18]. And then we carry out the same process todetermine the single parameter α. The parametersα ¼ ð7.51� 0.04Þ° and δ0 ¼ ð0.00� 0.01Þ° areobtained. The modulus of the corresponding PMNSmatrix is

jV4×4j ¼

0BBB@

0.826� 0.001 0.502� 0.001 0.161� 0.001 0.199� 0.002

0.492� 0.001 0.561� 0.001 0.659� 0.001 0.096� 0.001

0.241� 0.001 0.645� 0.001 0.724� 0.001 0.042� 0.001

0.131� 0.001 0.130� 0.001 0.128� 0.001 0.974� 0.001

1CCCA: ð15Þ

In this scenario, we assume that the mixing between thesterile neutrino and the different active neutrinos is non-distinctive. Our estimates are presented in Table I.Moreover, we find a new self-complementarity

ϑ012 þ ϑ023 þ ϑ013 þ α ≈ 90°, which is a bit different

from that self-complementarity relation given inRef. [13].As a test one would calculate the neutrino flavor number

that is determined to be 3 by the LEP data. Ignoring theneutrino masses, the neutrino number is

TABLE I. The values of jU00e4j2 and jU00

μ4j2 in this work and in references.

Ref. [19] Ref. [14] Ref. [22] Ref. [23] First scheme Second scheme

jU00e4j2 0.03–0.033 0.0185 0.027–0.036 0.0228 0.027� 0.004 0.040� 0.004

jU00μ4j2 0.0073–0.014 0.042 0.0084–0.021 � � � 0.077� 0.006 0.009� 0.002

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Nν ¼X4

ρ;σ¼1

ΓðZ → νρνσÞ=ΓðZ → ννÞ

¼X4

ρ;σ¼1

����X3

i¼1

ðV†ÞρiViσ

����2

; ð16Þ

where Viσ is the generalized PMNS matrix that is a 3 × 4matrix and not unitary. Our numerical result shows that inthis scenario,Nν is 3, which is fully consistent with the LEPmeasurement within a reasonable error tolerance. Thedenominator of the above equation ΓðZ → ννÞ stands forthe partial decay width of the Z boson into a neutrino paircalculated in the standard model (SM).

IV. SUMMARY

In this work we adopt the two quark-lepton comple-mentarity relations and a self-complementarity relationproposed in literature [7,9,11], which is supposed tooriginate from a higher symmetry and maintain whenthe energy scale runs down to the electroweak scale.Then the deviation of the determined values from the

measured PMNS matrix elements is attributed to theinvolvement of a sterile neutrino. The mixing of the sterileneutrino with the active ones results in the practical valuesof the PMNS matrix. Comparing with the data, we are ableto determine the mixing parameters.In this work, we choose two schemes: in the first scheme,

the sterile neutrino mixes with three different activeneutrinos by different parameters (i.e., α, β, and γ) areindependent parameters that are determined by fitting data;whereas in the second scheme, we let α ¼ β ¼ γ, so thatthere is only one parameter to describe the mixing. Thenumerical values are listed in Table I.It is noted that the previous estimates on the mixing

between the sterile neutrino and the active ones wereobtained by fitting the data. Instead, by our strategy, westart with the theoretical assumption: the complementarityand self-complementarity. The relevant mixing elementsobtained in previous literature are quite disperse, and theonly common point is that the sterile-active mixing is small,no matter how they are obtained.By the first scheme, our prediction on jU00

e4j2 is generallyconsistent with the results given by the authors ofRefs. [20,22] (see Table I), but the value of jU00

μ4j2 isslightly bigger. The compatibility of reactor antineutrinoanomaly was discussed in Ref. [24], and the mixingparameter sin22α ¼ 0.14� 0.08 was fixed whenΔm2

41 > 1.5 eV2. Our estimation on jU00e4j2 is consistent

with it also within a 2σ range.For the second scheme, the numbers look different, but

the trend and consistency degree with those given in theliterature are all within the present experimental errortolerance.

The theoretical uncertainties of our predictions originatefrom the measurement errors of the CKM matrix elementsthat are relatively small thanks to many years of hard work.On the contrary the experimental errors for measuring thePMNS matrix elements are larger. Thus, our predictions onthe mixing between sterile and active neutrinos and thatobtained by others are still consistent with each otherwithin 1–2σ ranges.Recently the Daya Bay Collaboration reports its new

data [25] on the mixing between the sterile neutrino andactive neutrinos, but the errors are still too large to make aconclusive judgment on the validity of our theory yet. Thefuture improved measurement may further narrow down thedata ranges, so that we can test any theoretical ansatz andget a better understanding of neutrinos.

ACKNOWLEDGMENTS

This work is supported by the National NaturalScience Foundation of China (NNSFC) under ContractsNo. 11375128 and No. 11135009. We greatly benefit fromProfessor Lam’s lecture given in Nankai University severalyears ago.

APPENDIX: The relevant matrices in4 × 4 PMNS matrix

R23¼

0BBB@

1 0 0 0

0 C23 S23 0

0 −S23 C23 0

0 0 0 1

1CCCA; R13¼

0BBB@

C13 0 S13 0

0 1 0 0

−S13 0 C13 0

0 0 0 1

1CCCA;

R12¼

0BBB@

C12 S12 0 0

−S12 C12 0 0

0 0 1 0

0 0 0 1

1CCCA; ðA1Þ

R14 ¼

0BBB@

Cα 0 0 Sα0 1 0 0

0 0 1 0

−Sα 0 0 Cα

1CCCA; R24 ¼

0BBB@

1 0 0 0

0 Cβ 0 Sβ0 0 1 0

0 −Sβ 0 Cβ

1CCCA;

R34 ¼

0BBB@

1 0 0 0

0 1 0 0

0 0 Cγ Sγ0 0 −Sγ Cγ

1CCCA; ðA2Þ

ϕ ¼

0BBB@

1 0 0 0

0 eiδ0

0 0

0 0 1 0

0 0 0 1

1CCCA; ðA3Þ

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