On neutrino mixing in matter and CP and T violation effects in neutrino oscillations

Contents lists available atScienceDirect

Physics

Letters

B

www.elsevier.com/locate/physletb

On

neutrino

mixing

in

matter

and

CP

and

T

violation

effects

in neutrino

oscillations

S.T. Petcov

a

,

b

,

1

,

Ye-Ling Zhou

c

,

∗

a_{SISSA/INFN,ViaBonomea265,34136Trieste,Italy}

b_{KavliIPMU(WPI),TheUniversityofTokyo,Kashiwa,Chiba277-8583,Japan}

c_{InstituteforParticlePhysicsPhenomenology,DepartmentofPhysics,DurhamUniversity,DurhamDH13LE,UnitedKingdom}

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory: Received14July2018

Receivedinrevisedform9August2018 Accepted17August2018

Availableonline20August2018 Editor:A.Ringwald

Aspects of 3-neutrino mixing and oscillations in vacuum and in matter with constant density are investigatedworkingwitharealformoftheneutrinoHamiltonian.Wefindthe(approximate)equalities

θm

23= θ23 andδm= δ,θ23 (θ23m)andδ(δm)beingrespectivelytheatmosphericneutrinomixingangleand

the DiracCPviolation phaseinvacuum (inmatter) ofthe neutrinomixing matrix,whichare shown torepresent excellentapproximationsforthe conditionsofthe T2K(T2HK),T2HKK,NOνAand DUNE neutrinooscillationexperiments.Anewderivationoftheknownrelationsin 2θm

23sinδm=sin 2θ23sinδis

presentedanditisusedtoobtainacorrelationbetweentheshiftsofθ23andδduetothemattereffect.

AderivationoftherelationbetweentherephasinginvariantswhichdeterminethemagnitudeofCPand T violatingeffects in 3-flavourneutrino oscillations invacuum, JCP, and ofthe T violatingeffects in

matterwithconstantdensity,Jm

T≡Jm,reportedin[1] withoutaproof,ispresented.Itisshownthatthe

function

F which

appearsinthisrelation, Jm=JCPF , andwhoseexplicitformwasgivenin[1],coincides

withthefunctionF in ˜ thesimilarrelation Jm_=J

CPF derived ˜ in[2],although

F and

F are ˜ expressedin

termsofdifferentsetsofneutrinomassandmixingparametersandhavecompletelydifferentforms. ©2018PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense

(http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP3_.

1. Introductionandpreliminaryremarks

It was shownin 1988 in ref. [1] that in the case of what is currentlyreferred toasthereference3-neutrinomixing(see,e.g., [3]),themagnitudeoftheCPandTviolating(Tviolating)effectsin neutrinooscillationsinvacuum(inmatter withconstantdensity) arecontrolledbytherephasinginvariant JCP( JmT

≡

Jm)associated withtheDiracCPviolationphasepresentinthePontecorvo,Maki, NakagawaandSakata(PMNS)[4,5] neutrinomixingmatrix:

JCP

(

Jm

)

=

Im



U_e2(m)

 

U_μ(m₃)

 

U_e3(m)



∗



U(_μm₂)



∗



,

(1)

whereU_li(m),l

=

e

,

μ

,

τ

, i

=

1,2,3,arethe elements ofthePMNS matrixinvacuum(inmatter) U(m)_{.The CPviolatingasymmetries}

inthecaseofneutrinooscillationsinvacuum,forexample,

*

Correspondingauthor.

E-mailaddress:ye-ling.zhou@durham.ac.uk(Y.-L. Zhou).

1 _{Alsoat:InstituteofNuclearResearchandNuclearEnergy,BulgarianAcademyof} Sciences,1784Sofia,Bulgaria.

A(_CPvacl,l)

=

Pvac

(

ν

l

→

ν

l

)

−

Pvac

(

ν

¯

l

→ ¯

ν

l) ,

l

=

l and l

,

l

=

e

,

μ

,

τ

,

(2)

Pvac

(

ν

l

→

ν

l

)

andPvac

(

ν

¯

l

→ ¯

ν

l

)

beingtheprobabilitiesof respec-tively

ν

l

→

ν

l and

ν

¯

l

→ ¯

ν

l oscillations, were shown to be given by [1]:

A(_CPvace,μ)

=

A(_CPvacμ,τ)

= −

A(_CPvace,τ)

=

4 JCP



vacosc

,

(3) with



vac_osc

=

sin





m2₂₁L 2E



+

sin





m2₃₂L 2E



+

sin





m2₁₃L 2E



,

(4)

where



m2_{i j}

=

m2_i

−

m2_j, i

=

j, mi, i

=

1,2,3, is the mass of the neutrino

ν

i with definite massin vacuum, E is the neutrino en-ergyandL isthedistancetravelledbytheneutrinos.In[1] similar resultswereshowntobevalidfortheT-violating asymmetriesin oscillations in vacuum (in matter), A_Tvac(l,l)_(m)

=

Pvac(m)

₍

_ν

l

→

ν

l

)

−

Pvac(m)

(

ν

l

→

ν

l

):

https://doi.org/10.1016/j.physletb.2018.08.025

A_Tvac(μ,e)_(m)

=

A_Tvac(τ,μ₍)_m)

= −

A_Tvac(e,τ)_(m)

=

4 J_CP(m₍)_T)



vacosc(m)

,

(5) where



mosc has the sameform as



vacosc in eq. (4) with



m2i j re-placedby masssplittinginthe matter



M2

i j

=

M2i

−

M2j,andMi, i

=

1,2,3,areneutrinomass-eigenvaluesinthematter.Invacuum the T violating asymmetries in antineutrino oscillations, A

¯

(_Tvacl,l)

=

Pvac

(

ν

¯

l

→ ¯

ν

l

)

−

Pvac

(

ν

¯

l

→ ¯

ν

l

),

arerelatedtothoseinneutrino os-cillationsowingtotheCPTinvariance: A

¯

_Tvac(l,l)

= −

A(_Tvacl,l).Inordinary matter (Earth, Sun)2 the presence of matter causes CP and CPT violating effects in neutrino oscillations [6] and

| ¯

A_Tm(l,l)

|

= |

A(_Tml,l)

|

. However,inordinarymatterwithconstantdensityorwithdensity profile which is symmetric relative to the middlepoint, likethe matter of theEarth, the matter effects preserve the Tsymmetry anddonotgenerateTviolatingeffectsinneutrinooscillations[1]. Thus,Tviolatingeffectsintheflavourneutrinooscillations taking place when the neutrinos traverse, e.g., the Earth mantleor the Earthcorecanbecausedinthecaseof3-neutrinomixingonlyby theDiracphaseinthePMNSmatrix.

The JCP-factorin the expressions for A(l,l ₎

CP(T)vac,l

=

l, is anal-ogous tothe rephasing invariant associatedwiththe Diracphase in the Cabibbo–Kobayashi–Maskawa (CKM) quark mixing matrix, introduced in [7]. In the standard parametrization of the PMNS mixingmatrix(see,e.g.,[3])ithastheform:

JCP

=

1

8 cos

θ

13sin 2

θ

12sin 2

θ

23sin 2

θ

13sin

δ ,

(6)

where

θ

12,

θ

23and

θ

13arethesolar,atmosphericandreactor neu-trino mixing angles and

δ

is the Dirac CP violation phase. The expression for the JCP-factor is the same in the parametrisation ofthePMNSmatrixUPMNS

≡

U employedin[1]:

U

=

R23

(θ

23

)

P33

(δ)

R13

(θ

13

)

R12

(θ

12

) ,

(7) where R23

(θ

23

)

=

⎛

⎝

10 cos0

θ

23 sin0

θ

23 0

−

sin

θ

23 cos

θ

23

⎞

⎠ ,

P33

(δ)

=

diag

(

1

,

1

,

eiδ

) ,

(8) and R13

(θ

13

)

=

⎛

⎝

cos0

θ

13 01 sin0

θ

13

−

sin

θ

13 0 cos

θ

13

⎞

⎠ ,

R12

(θ

12

)

=

⎛

⎝

₋

cossin

θ

θ

1212 cossin

θ

θ

1212 00

0 0 1

⎞

⎠ .

(9)

The expressionof thePMNS matrix inthe standard parametrisa-tion Usp, is related to the expression in the parametrisation in eq. (7) asfollows:Usp

=

U P∗₃₃

(δ).

Ineq. (7) thetwoCPviolation(CPV)Majoranaphasespresentin

UPMNS inthecaseofmassiveMajorananeutrinos[8] wereomitted since,aswas shownin[8,6],the probabilitiesofflavour neutrino oscillationsofinterest forthestudyperformedin[1] and forthe presentstudy, donot depend on the Majoranaphases. Thus, the resultspresentedin[1] andthenewresultsderivedinthepresent articlearevalidforbothDiracandMajorananeutrinoswith defi-nitemassesinvacuum.

2 _{By“ordinary”wemeanmatterwhichdoesnotcontainantiprotons,antineutrons} andpositrons.

In ref. [1] the following relation between the rephasing in-variants invacuumandinmatter withconstantdensity, JCP and

Jm_T

≡

Jm,hasbeenreported:

Jm

=

JCPF

(θ

12

, θ

13

,



m221

,



m231

,

A

) ,

(10) where A

=

2E

√

2 GFNe isthematterterm[9–11], GFandNe be-ingrespectivelytheFermiconstantandtheelectronnumber den-sityofmatter.ThefunctionF ineq. (10) wasgiveninthefollowing explicitformin[1]: F

=

F1 F2F3 D12D13D23D32

,

(11) where Di j

≡

m2i

−

M2j

,

i

,

j

=

1

,

2

,

3

,

(12) F1

=

D12D13D23D32

+

A

D13D23



D32

− 

m231

|

Ue3

|

2



+

D12D32



D23

− 

m221

|

Ue2

|

2



+

A2

|

Ue1

|

2D32D23

+ |

Ue2

|

2D32D13

+ |

Ue3

|

2D12D23



,

(13) F2

= |

Ue1

|

2

(

D12

+

A

)

2D232

+ |

Ue2

|

2D212D232

+ |

Ue3

|

2D212

(

D32

+

A

)

2

−

A2

|

Ue1

|

2

|

Ue3

|

2

(

m231

)

2

,

(14) F3

= |

Ue1

|

2

(

D13

+

A

)

2D223

+ |

Ue3

|

2D213D223

+ |

Ue2

|

2D213

(

D23

+

A

)

2

−

A2

|

Ue1

|

2

|

Ue2

|

2

(

m221

)

2

.

(15) As was noticed in [1], the function F3 can formally be ob-tained from the function F2 by interchanging m22 and m23, M22 and M2

3, and

|

Ue2

|

2 and

|

Ue3

|

2. In the parametrisation (7) used in [1] and, thus in eqs. (13)–(15), Ue1, Ue2 and Ue3 are real quantities: Ue1

=

c12c13,Ue2

=

s12c13 and Ue3

=

s13, whereci j

≡

cos

θ

i j andsi j

≡

sin

θ

i j.Thus,

|

Uei

|

2

=

Uei2,i

=

1,2,3.The function F

(θ

12

,

θ

13

,



m2₂₁

,



m2₃₁

,

A

)

asdefinedby eqs. (11)–(15), depends, in particular,on thedifferencesbetweenthe squaresof the neu-trinomassesinvacuumandinmatter, Di j

=

m2i

−

M2j,i

=

j. How-ever,asitfollowsfromtheformoftheHamiltonianoftheneutrino system in matter with constant density, whose eigenvalues are

M2_j

/(2E)

(seefurther),aswellasfromtheexplicitanalytic expres-sions forM2

j derived in[10], themasssquareddifferences Di j of interest arefunctionsof

θ

12,

θ

13,



m2₂₁,



m2₃₁ and A anddo not dependon

θ

23and

δ.

Asaconsequence,thefunction F ineq. (10) isindependenton

θ

23and

δ

[1]:F

=

F

(θ

12

,

θ

13

,



m221

,



m231

,

A

).

In deriving the relation (10), the following parametrisation of theneutrinomixingmatrixinmatterUm _wasused:

Um

=

Q R23

(θ

23m

)

P33

(δ

m

)

R13

(θ

13m

)

R12

(θ

12m

) ,

Q

=

diag

(

1

,

eiβ2

_,

_eiβ3

_{) ,}

₍₁₆₎

where

θ

₂₃m,

θ

₁₃m,

θ

₁₂m,

δ

m are the neutrino mixing angles and the Dirac CPV phase in matter and the Majorana CPV phases were omitted.Thephases

β

2and

β

3 inthematrixQ areunphysicaland donotplayanyroleinthederivationofrelation(10).Theyensure that thematrixUm canbecastintheformgivenineq. (16) [12] (seealso[13]).Obviously,theparametrisationofUmineq. (16) is analogoustotheparametrisation(7) oftheneutrinomixingmatrix invacuum.

Itfollowsfromeqs. (11)–(15) that[1] inthecaseofoscillations invacuum,i.e.,forNe

=

0 ( A

=

0),onehas

F

(θ

12

, θ

13

,

m221

,



m231

,

0

)

=

1

,

(17)

andthat F issymmetricwithrespecttotheinterchangeofm2 2 and

m2₃,M2₂ andM2₃ andof

|

Ue2

|

2 and

|

Ue3

|

2. The relation (10) between Jm _{and J}

CP implies, in particular, thatwecanhave Jm

=

0 onlyif JCP

=

0,i.e.,Tviolationeffectscan bepresentinneutrinooscillationstakingplaceinmatterwith con-stant densityor densitydistributed symmetricallyrelative to the middlepoint(like intheEarth) onlyifCPandTviolationeffects arepresentinneutrinooscillationstakingplaceinvacuum.Itwas shownalsoin[1] thatthepresenceofmattercan enhance some-what

|

Jm

_|

_{withrespecttoits vacuumvalue}

_|

_J

CP

|

: intheexample considered in [1] the enhancement was by a factor of 3. Taking thebestfitvaluesof neutrinooscillationparametersforneutrino massspectrumwithnormalordering(invertedordering)3obtained intheglobalanalysisin[14],

θ

12

=

33

.

62◦

, θ

23

=

47

.

2◦

(

48

.

1◦

) ,

sin2

θ

13

=

8

.

54◦

(

8

.

58◦

) ,

δ

=

234◦

(

278◦

) ,



m₂₁2

=

7

.

4

×

10−5eV2

,



m2₃₁

=

2

.

494

×

10−3eV2

(

m2₃₂

= −

2

.

465

×

10−3eV2

) ,

(18)

onealways hasfor theratio

|

Jm

_/

_J

CP

|

<

1.2 [15].This result per-sists even if we fix

δ

to its best fit value and vary the other neutrino oscillation parameters in their 3

σ

allowed ranges de-termined in [14]. Relaxing arbitrarily the 3

σ

experimental con-straintsontheallowedrangesof



m2₂₁and



m2₃₁,wefindthat in-deedthemaximalenhancementfactor

|

Jm

_/

_J

CP

|

is3.6forneutrino massspectrum with normalordering (NO) and2.9for spectrum withinvertedordering(IO). Inboth cases,themaximal enhance-ment corresponds to Jm _{reaching its theoretical maximal value} max(

|

Jm

|)

=

1/(6

√

3).

In1991in [2] arelation similar tothat givenin eq. (10) was obtained:

Jm

=

JCPF

˜

.

(19)

Thefunction F was

˜

giveninthefollowingform:

˜

F

=



m 2 12



m223



m231



M2 12



M223



M312

,

(20) where



M2 i j

=

M2i

−

M2j.

Therelation(10) betweentherephasinginvariants Jm _{and J} CP waspresentedin[1] withouta proof.In thepresentarticle,after discussingcertainaspectsofneutrinomixinginmatter,weprovide aderivation oftherelation(10).Further, weshow that the func-tion F ineq. (10),asdefinedineqs. (13)–(15), coincideswiththe functionF in

˜

therelation(19) obtainedin[2],

F

= ˜

F

,

(21)

i.e.,thatthefunction F is justanotherrepresentationofthe func-tionF .

˜

2. Onthe3-neutrinomixinginmatter

In[1] theanalysis was performed starting withthe following Hamiltonianof the neutrino systemin matter diagonalised with thehelpoftheneutrinomixingmatrixinmatterUm[6]:

3 _{Foradiscussionofthedifferentpossibletypesofneutrinomassspectrumsee,} e.g.,[3]. 1 2EU

⎡

⎣

⎛

⎝

m 2 1 0 0 0 m2₂ 0 0 0 m2₃

⎞

⎠ +

U†

⎛

⎝

A0 0 00 0 0 0 0

⎞

⎠

U

⎤

⎦

U†

=

1 2EU m

⎛

⎝

M 2 1 0 0 0 M2 2 0 0 0 M2₃

⎞

⎠ (

Um

)

†

.

(22)

It follows from the preceding equation4 _{that the Hamiltonian of}

theneutrinosystem,

H

=

1 2E

⎛

⎝

m 2 1 0 0 0 m2₂ 0 0 0 m2₃

⎞

⎠ +

1 2EU †

⎛

⎝

A0 0 00 0 0 0 0

⎞

⎠

U (23)

=

1 2E

⎛

⎝

m 2 1

+

A

|

Ue1

|

2 AU∗e1Ue2 AU∗e1Ue3 AU_e2∗Ue1 m22

+

A

|

Ue2

|

2 AU∗e2Ue3 AU_e3∗Ue1 AU∗_e3Ue2 m2₃

+

A

|

Ue3

|

2

⎞

⎠

(24)

is diagonalised by the matrix U†_Um _{and its eigenvalues are} M2

i

/(2E),

i

=

1,2,3.Intheparametrisation(7) ofthePMNSmatrix Ue1, Ue2 and Ue3 are real quantities: Ue1

=

c12c13, Ue2

=

s12c13 and Ue3

=

s13, where ci j

≡

cos

θ

i j and si j

≡

sin

θ

i j. As a con-sequence, the Hamiltonian H is a real symmetric matrix.5 _This

impliesthatthematrixU†Um,whichdiagonalises H ,isareal or-thogonalmatrix:

U†Um

=

O

,

O∗

=

O

,

OTO

=

O OT

=

diag

(

1

,

1

,

1

) .

(25)

Since H does not depend on

θ

23 and

δ,

O

=

U†Um should not dependon

θ

23and

δ

either.ThefactthatthematrixO ineq. (25) isarealorthogonalmatriximpliesthatintheparametrisations(7) and(16) ofthePMNSmatrixinvacuumandinmatter,thematrix

˜

O

=

R13

(θ

13

)

R12

(θ

12

)

O R12T

(θ

12m

)

RT13

(θ

13m

)

=

P∗₃₃

(δ)

RT₂₃

(θ

23

)

Q R23

(θ

23m

)

P33

(δ

m

)

(26)

=



_{1 0 0} 0 cm 23c23eiβ2+sm23s23eiβ3 sm23c23ei(β2+δ m₎ −cm 23s23ei(β3+δ m₎ 0 s23cm23ei(β2−δ)−c23s23mei(β3−δ) sm23s23ei(β2−δ+δ m₎ +cm 23c23ei(β3−δ+δ m₎



,

(27)

isarealorthogonalmatrix.

The requirement of reality of the nondiagonal elements of O

˜

leadstotheconditions:

cos

θ

23sin

θ

23m sin

(β

3

− δ) =

sin

θ

23cos

θ

23m sin

(β

2

− δ) ,

cos

θ

23sin

θ

23m sin

(β

2

+ δ

m

)

=

sin

θ

23cos

θ

23m sin

(β

3

+ δ

m

) ,

(28) whichimply,inparticular:

cos

(

2

β

3

+ δ

m

− δ) =

cos

(

2

β

2

+ δ

m

− δ) .

(29) Thelastconditionhastwosolutions:

β

3

= β

2

+

k

π

,

k

=

0

,

1

,

2

, ... ,

(30)

β

2

+ β

3

= δ − δ

m

+

k

π

,

k

=

0

,

1

,

2

, ... .

(31)

4 _{TheCPVMajoranaphasesα}

21andα31,enterintotheexpressionforthePMNS matrixinvacuumthroughthediagonalmatrix P=diag(1, eiα21/2, _eiα31/2_)[8,16]):

UPMNS=U P .Itfollowsfromtheexpressioninthelefthandsideofeq. (22) that theHamiltonianofneutrinosysteminmatter,andthusthe3-flavourneutrino os-cillationsinmatter,donotdependontheMajoranaphases[6].

5 _{ReplacingthematrixU withU}sp_{=U P}∗

33(δ)ineq. (22),itiseasytoconvince oneselfthattheHamiltonianH hastheformgivenineq. (24) alsointhestandard parametrisationofthePMNSmatrixwithUe3replacedby|Ue3|=s13.

TherequirementofrealityofthediagonalelementsofO leads

˜

to:

cos

θ

23cos

θ

23m sin

β

2

= −

sin

θ

23sin

θ

23m sin

β

3

,

cos

θ

23cos

θ

23m sin

(β

3

− δ + δ

m

)

= −

sin

θ

23sin

θ

23m sin

(β

2

− δ + δ

m

) .

(32) Theseconditionsalsolead,inparticular,totheconstraintgivenin eq. (29) andtothesolutions(30) and(31).Itshouldbe clearthat satisfyingtheconstraint(30) or(31) isnot enough toensurethe realityofthematrix O .

˜

Consider first the consequences of the constraint in eq. (30). Requiringinadditionthat thedeterminantof O is

˜

arealquantity implies:

2

β

2

= δ − δ

m

+

k

π

,

k

=

0

,

1

,

2

, ...

(33) Theconstraintineq. (30) fork

=

0,forexample,andtheconditions ofrealityoftheelements

( ˜

O

)

23(32)and

( ˜

O

)

22(33)of O lead

˜

to:

sin

(θ

23

− θ

23m

)

sin

(β

2

− δ) =

0

,

sin

(θ

23

− θ

23m

)

sin

(β

2

+ δ

m

)

=

0

.

(34) cos

(θ

23

− θ

23m

)

sin

β

2

=

0

,

cos

(θ

23

− θ

23m

)

sin

(β

2

− δ + δ

m

)

=

0

.

(35)

Asaconsequenceofeq. (33) thesecondconditionsineqs. (34) and (35) areequivalenttothefirstconditionsineqs. (34) and(35).The constraint(30) fork

=

0 andtheconditions(34) and (35) canbe simultaneouslysatisfiedifthefollowingrelationshold6_:

θ

₂₃m

= θ

23

,

0

< θ

23

, θ

23m

≤

π

/

2

,

(36)

δ

m

= δ +

k

π

,

k

=

0

,

1

,

2

,

(37)

β

2

=

q

π

,

q

=

0

,

1

,

2

, ... .

(38)

Numericalresultsonthedependenceof

θ

₂₃m,

δ

m,

θ

m₂₃and

δ

monthe matterpotential A,

θ

m₂₃ and

δ

m beingthe corresponding antineu-trinomixingangleandCPviolationphase,showthattherelations (36) and (37) cannot be exact. These relations are not the true solutions ofthereality conditions ofthe matrix O since

˜

they do not fully guaranteethe reality of O as

˜

givenin eq. (27). Only if both

θ

23

=

45◦ and

δ

= ±

π

/2 hold

invacuum, therelations (36) and(37) areexact and are not violated by theeffects of matter [17].However,they arefulfilledwithextremelyhighprecision for themixingofneutrinos(antineutrinos)inmatterinthecaseofIO (NO) neutrino mass spectrum. We find that in thiscase for any

A

/

m2

21andthebestfitvaluesoftheneutrinooscillation param-etersquotedineq. (18) wehave:





(−)

θ

m₂₃

θ

23

−

1



 

0

.

0004

(

0

.

0015

) ,

(39)





(−)

δ

m

δ

−

1



 

0

.

0001

(

0

.

00006

) .

(40)

Forthemixingofneutrinos(antineutrinos)inmatterandspectrum with normal (inverted) ordering, eqs. (36) and (37) are fulfilled alsowithextremelyhighprecisionfor A

/

m2₂₁

<

30:

6 _{An alternative solution to the discussed constraints is θ}m

23= θ23, β2=qπ, q=0, 1, 2, ...,

δ

=kπ,k=0, 1, 2,

δ

m_{= ˜k}_π,k˜=0, 1, 2.ItcorrespondstoCP(T) conservingvaluesof

δ

(δm).





(−)

θ

m₂₃

θ

23

−

1



 

0

.

006

(

0

.

0015

) ,

(41)





(−)

δ

m

δ

−

1



 

0

.

0003

(

0

.

001

) .

(42)

For7 A

/

m2₂₁



30 andmixingofneutrinos(antineutrinos)in mat-terandNO(IO)neutrinomassspectrumwehave:





(−)

θ

m₂₃

θ

23

−

1



 

0

.

07

(

0

.

016

) ,

(43)





(−)

δ

m

δ

−

1



 

0

.

001

(

0

.

004

) .

(44)

Setting

δ

toitsbestfitvalue givenineq. (18) andvaryingthe other neutrinooscillation parameters intheir 3

σ

allowed ranges determined in[14] doesnotchangesignificantlytheresultsquoted in eqs. (40)–(44). Indeed, formixing of neutrinos(antineutrinos) inmatterinthecaseofIO(NO)neutrinomassspectrumandany

A

/

m2₂₁wefindthat

|

(−)

θ

m₂₃

/θ

23

−

1

|



0.0005

(0.002)

and

|

(−)

δ

m

/δ

−

1

|



0.0002

(0.0002).

Inthecaseofmixingofneutrinos (antineu-trinos)inmatterandNO(IO)spectrumand A

/

m2

21

<

30 weget

|

(−)

θ

m₂₃

/θ

23

−

1

|



0.013

(0.003)

and

|

(−)

δ

m

/δ

−

1

|



0.0013

(0.003),

whileforA

/

m2₂₁



30 weobtain

|

(−)

θ

m₂₃

/θ

23

−

1

|



0.09

(0.02)

and

|

(−)

δ

m

/δ

−

1

|



0.01

(0.01).

TheseresultsareillustratedinFigs.1and

2

(3and

4

)wherethe ratios

θ

₂₃m

/θ

23 and

δ

m

/δ

(the ratios

θ

m23

/θ

23 and

δ

m

/δ)

areshown as functionsof A

/

m2

21 in the case of mixingof neutrinos (an-tineutrinos)andNO(leftpanel)andIO(rightpanel)neutrinomass spectrum. We usedthe bestfit valuesof neutrinooscillation pa-rameters



m2₃₁,



m2₂₁,

θ

12 and

θ

13 from [14] and the analytic expressionsforM2

i,i

=

1,2,3,from[12].

The approximate ranges of values of A

/

m2

21 relevant for the T2K (T2HK) [19], T2HKK [20], NO

ν

A [21] and DUNE [22] long baseline neutrino oscillation experiments read, respectively:

[

0.266,2.66

]

,

[

0.306,3.06

]

,

[

2.90,8.70

]

and

[

3.02,12.10

]

. In ob-taining theseranges we usedthe bestfit value of



m2

21

=

7.4

×

10−5 _eV2 _{andtook intoaccount i) that A}

₌

_7.56

_×

₁₀−5 _eV2

(

ρ

/

g/cm3

₎

₍

_E

_/GeV),

_where

_ρ

_{isthematterdensity,ii)thatthemean} Earth density along the trajectories of the neutrinos in the T2K (T2HK), T2HKK, NO

ν

A and DUNE long baseline neutrino oscilla-tion experiments respectively is 2.60, 3.00, 2.84and 2.96 g/cm3_, andiii) that intheseexperiments beamsofneutrinos with ener-gies

∼ (

0.1–1.0) GeV(T2K, T2HK, T2HKK),

∼ (

1–3) GeV (NO

ν

A) and

∼ (

1–4)GeV(DUNE)arebeing,orplannedtobe,used.Atthe peak neutrino energies at T2K (T2HK), T2HKK, NO

ν

A andDUNE experimentsofrespectively0.6 GeV,0.6 GeV,2.0 GeVand2.6 GeV we have A

/

m2

21

=

1.59, 1.84, 5.80 and 7.86. Taking a wider neutrino energy interval for, e.g., NO

ν

A and DUNE experiments of

[

1.0,8.0

]

GeV, we get for the corresponding A

/

m2₂₁ ranges:

[

2.90,23.21

]

and

[

3.02,24.20

]

. For all the intervals of values of

A

/

m2

21 quoted above, which are relevant for the T2K (T2HK), T2HKK,NO

ν

AandDUNEexperiments,theequalities(36) and(37) areexcellentapproximations.

Considernexttheimplicationsofthesecondcondition(31) re-lated to the requirement of reality of the matrix O .

˜

As can be easilyshown,thisconditionalonei) ensurestherealityofdet( ˜O

),

Fig. 1. Theratio

θ

m

23/θ23asafunctionofA/m221inthecaseofmixingofneutrinosandNO(leftpanel)andIO(rightpanel)neutrinomassspectrum.Seetextforfurther details.

Fig. 2. Theratio

δ

m_{/δasafunctionofA/m}2

21inthecaseofmixingofneutrinosandNO(leftpanel)andIO(rightpanel)neutrinomassspectrum.Seetextforfurther details.

Fig. 3. The same as in Fig.1but for the ratioθm

23/θ23. See text for further details.

andii)makesidenticalthetwoconditionsineq. (28) andthetwo conditionsin eq. (32). Thus, after using condition (31) there are stilltwo independent conditionstobe satisfiedto ensurethe re-ality of the matrix O .

˜

We will derive next a condition that can substituteone ofthe requiredtwo conditions.The second condi-tionthencanbe eitherthecondition ineq. (28) or thecondition ineq. (32) (aftereq. (31) hasbeenused).

Thecondition of orthogonalityof O ,

˜

O

˜

( ˜

O

)

T

=

diag(1,1,1), as canbeshown,leadstothefollowingadditionalconstraints:

(

cm₂₃

)

2sin

(

2

β

2

− δ) + (

sm23

)

2sin

(

2

β

2

− δ +

2

δ

m

)

= −

cm23sm23

c23s23

cos 2

θ

23sin

δ

m

,

(45)

sin 2

θ

23sin 2

θ

23m sin

δ

sin

δ

m

+ (

cm₂₃

)

2cos

(

2

β

2

)

+ (

sm23

)

2cos

(

2

β

2

+

2

δ

m

)

=

1

−

2 s2₂₃sin

δ

[(

cm₂₃

)

2sin

(

2

β

2

− δ)

Fig. 4. The same as Fig.2but for the ratioδm_{/δ. See text for further details.}

Fig. 5. Theratiosin 2θm

23sinδm/(sin 2θ23sinδ),eq. (49),formixingofneutrinosasafunctionofA/m221inthecasesofNO(leftpanel)andIO(rightpanel)neutrinomass spectra.Seetextforfurtherdetails.

−

sin 2

θ

23sin 2

θ

23m cos

δ

sin

δ

m

+ (

cm23

)

2sin

(

2

β

2

)

+ (

sm₂₃

)

2sin

(

2

β

2

+

2

δ

m

)

=

2 s2₂₃cos

δ

[(

cm₂₃

)

2sin

(

2

β

2

− δ)

+ (

sm₂₃

)

2sin

(

2

β

2

− δ +

2

δ

m

)

] ,

(47) where we have used the relation in eq. (31). Conditions (45), (46) and (47) follow from the requirements

( ˜

O

( ˜

O

)

T

)

23(32)

=

0,

Re(( ˜O

( ˜

O

)

T

)

22

)

=

1 andIm(( ˜O

( ˜

O

)

T

)

22

)

=

0,respectively.Replacing

(

cm₂₃

)

2sin(2β2

− δ)

+ (

sm23

)

2sin(2β2

− δ +

2δm

)

ineqs. (46) and(47) withthe righthand side ofeq. (45), after certain simplealgebra leadstotheequality:

sin 2

θ

₂₃m sin

δ

m

=

sin 2

θ

23sin

δ .

(48)

Thisresultwas derived in [23] (seealso [13])usingthe parame-terisations (7) and(16) introducedin[1] butemployingadifferent method.8 The equality(48) implies that theproduct sin 2θ23sin

δ

does not depend on the matter potential, i.e., is the same for neutrino oscillations takingplace in vacuum andin matter with constant density.It is validforneutrino andantineutrino mixing

8 _{The method employed in [23] is based on the observation [24] that the} parametrisation(7) allowstofactor outthe part R23(θ23)P33(δ)intheneutrino mixing matrix in matter. In this case one works with the Hamiltonian Hˆ = P∗₃₃(δ)RT

23(θ23)U H U†R23(θ23)P33,whichisalsoarealsymmetricmatrix,whereH isgivenineq. (24).

inmatterindependentlyofthetype ofspectrum neutrinomasses obey – withNO or IO. From eq. (10) using the parametrisations definedineqs. (7) and(16) wefind:

sin 2

θ

₂₃msin

δ

m sin 2

θ

23sin

δ

=

F cos

θ

13sin 2

θ

12sin 2

θ

13 cos

θ

₁₃m sin 2

θ

₁₂m sin 2

θ

₁₃m

=

F

Ue1Ue2Ue3

Um_e1Um_e2Um_e3

.

(49)

Fromthisresultandeq. (48) weobtainyetanotherequivalent rep-resentationofthefunction F

(θ

12

,

θ

13

,



m221

,



m231

,

A

):

F

=

U

m e1Ue2mUme3

Ue1Ue2Ue3

.

(50)

The ratio given in eq. (49) is shown graphically in Fig. 5 for mixingofneutrinosasafunctionofA

/

m2₂₁forthebestfitvalues ofneutrinooscillationparameters



m2

31,



m221,

θ

12 and

θ

13from [14] andtheanalyticexpressionsforM_i2,i

=

1,2,3,from[12].The numericalresultpresentedinFig.5,aswehaveverifiedandcould be expected, is validnot only for best fit values of the relevant neutrinooscillationparameters,butindeedholdsforanyvaluesof these parameters, varied intheir respectivephysical regions. The sameresultisvalidformixingofantineutrinos.Thus,theequality (48) isexact.It holdsalsointhestandardparametrisationsofthe PMNS matrix (see footnote 4). In thiscase the ratio Ue3

/

Ume3 in eqs. (49) and(50) hastobereplacedby

|

Ue3

|/|

Um_e3

|

.

It followsfromeq. (48) thatforthevaluesofsin 2θ23

=

1 and

δ

=

3

π

/2,

whichare perfectlycompatiblewith theexisting data,

wehavesin 2θ₂₃m sin

δ

m

= −

1.Thisinturnimpliessin 2θ₂₃m

=

1 and sin

δ

m

= −

1,i.e., thevacuum valuesof

θ

23

=

π

/4 and

δ

=

3

π

/2

arenotmodifiedbythepresenceofmatter[17].

Giventhefactthat,aswehaveseen,thecorrectionsof

θ

23and

δ

dueto thematter effects are small, therelation (48) allows to relatethemattercorrectionto

θ

23,

23

(

A

/

m221

),

withthematter correctionto

δ,

δ

(

A

/

m2

21

).

Working toleading orderin

23



1 and

δ



1 weget fromeq. (48) using

θ

₂₃m

= θ

23

+

23 and

δ

m

=

δ

+

δ:

δ

(

A

/

m221

)

cos

δ ∼

= −

2

23

(

A

/

m221

)

cos 2

θ

23

sin 2

θ

23

sin

δ .

(51)

Thus, for

θ

23

=

π

/4 and

δ

=

3

π

/2,

π

/2,

the leading order mat-tercorrection to

δ

vanishes, whilefor

δ

=

3

π

/2 (

π

/2)

and

θ

23

=

π

/4,

the leading order matter correction to

θ

23 vanishes. For

δ

=

q

π

/2,

q

=

0,1,2,3,4, and

θ

23

=

π

/4 the

signof

δ

coincides with (is opposite to) the sign of

23 provided cos 2θ23cot

δ <

0 (cos 2θ23cotδ >0).

3. Therelationbetween Jmand JCP Equation(22) canbecastintheform:

⎡

⎣

⎛

⎝

m 2 1 0 0 0 m2 2 0 0 0 m2₃

⎞

⎠ +

U†

⎛

⎝

A0 0 00 0 0 0 0

⎞

⎠

U

⎤

⎦ (

U†Um

)

= (

U†Um

)

⎛

⎝

M 2 1 0 0 0 M2 2 0 0 0 M2₃

⎞

⎠ .

(52)

One possible relatively simple way to derive the relation be-tween Jm_{and J}

CPgivenineq. (19) andreportedin[1] istoexploit thefactthatthecolumnmatrices

((

U†Um

)

1i

(

U†Um

)

2i

(

U†Um

)

3i

)

T areeigenvectors ofthe Hamiltonian H definedineq. (24), corre-spondingtotheeigenvalues M2

i

/(2E),

i

=

1,2,3.Usingthis obser-vationitispossibletoderivefromeq. (52) explicitexpressionsfor the elements of the neutrinomixing matrix in matter Um. They read: U_lim

=

1 Di

NiUli

−

A Uei



DjiU∗ekUlk

+

DkiU∗ejUlj



,

l

=

e

,

μ

,

τ

,

(53) where Ni

=

DjiDki

+

A



Dji

|

Uek

|

2

+

Dki

|

Uej

|

2



,

(54) D2_i

=

N_i2

+

A2

|

Uei

|

2



D2_ji

|

Uek

|

2

+

D2ki

|

Uej

|

2



_,

₍₅₅₎

withi

,

j

,

k

=

1,2,3,buti

=

j

=

k

=

i.Forthe elements ofUm _of interest,Um_e2,U_e3m,U_μm₂ andUm_μ3 wegetfromeqs. (53)–(55):

Um_e2

=

1 D2 Ue2D12D32

,

(56) Um_e3

=

1 D3 Ue3D13D23

,

(57) Um_μ2

=

1 D2



N2Uμ2

−

A Ue2



D12U∗e3Uμ3

+

D32Ue1∗ Uμ1



,

(58) and Um_μ3

=

1 D3



N3Uμ3

−

A Ue3



D13U∗e2Uμ2

+

D23U∗e1Uμ1



.

(59)

Thefunction D2₂,which,asitfollowsfromeqs. (1) and(56)–(59), entersintotheexpressionfor Jm_,isgivenby:

D2₂

=

N2₂

+

A2

|

Ue2

|

2



D2₁₂

|

Ue3

|

2

+

D232

|

Ue1

|

2



=

D2₁₂D2₃₂

+

2 A D12D32



D12

|

Ue3

|

2

+

D32

|

Ue1

|

2



+

A2

D2₁₂

|

Ue3

|

2



|

Ue3

|

2

+ |

Ue2

|

2



+

D2₃₂

|

Ue1

|

2



|

Ue1

|

2

+ |

Ue2

|

2



+

2 D12D32

|

Ue1

|

2

|

Ue3

|

2



=

D2₁₂D2₃₂

+

2 A D12D32



D12

|

Ue3

|

2

+

D32

|

Ue1

|

2



+

A2

D2₁₂

|

Ue3

|

2

+

D232

|

Ue1

|

2

− (

D32

−

D12

)

2

|

Ue1

|

2

|

Ue3

|

2



.

(60)

It iseasy tocheck thatexpression (60) for thefunction D2 2 coin-cideswithexpression(14) forthefunction F2,i.e.,thatwehave

D2₂

=

F2

.

(61)

OnecanshowinasimilarwaythatthefunctionD2₃ coincideswith thefunctionF3 givenineq. (15),i.e.,that

D2₃

=

F3

.

(62)

The calculation of the rephasing invariant in matter Jm in-volves,inparticular,theproductUm_μ3

(

U_μm₂

)

∗

(

U_e3m

)

∗Um_e2ofelements ofUm.Fromeqs. (56)–(59) wehave:

R

≡

D 2 2D23 D13D23D12D32 Im



(

Um_μ2

)

∗U_μm₃

(

Um_e3

)

∗Um_e2



Im



U∗_μ2Ue2Uμ3U_e3∗



(63)

=

1 Im



U_μ∗₂Ue2Uμ3U∗e3



Im



N∗₂U_μ∗₂Ue2

−

A

|

Ue2

|

2



D12Ue3Uμ∗3

+

D32Ue1Uμ∗1



×



N3Uμ3U∗e3

−

A

|

Ue3

|

2



D13Ue2∗ Uμ2

+

D23Ue1∗ Uμ1

 

.

(64)

Usingthefactthat

JCP

=

Im

(

U∗μ2Ue2Uμ3Ue3∗

)

=

Im

(

U∗μ3Ue3Uμ1Ue1∗

)

= −

Im

(

U∗_μ2Ue2Uμ1Ue1∗

) ,

(65)

the function R in eq. (64),after some algebra,can bebrought to theform: R

=

D13D23D12D32

+

A



D13D23



D32

− |

Ue3

|

2

(

D32

−

D12

)



+

D12D32



D23

− |

Ue2

|

2

(

D23

−

D13

)



+

A2



D23D32

|

Ue1

|

2

+

D13D32

|

Ue2

|

2

+

D12D23

|

Ue3

|

2



.

(66)

Equations(11),(10),(63) andtheequalities F2

=

D2₂ and F3

=

D2₃ provenabove,togetherwiththeequalities D32

−

D12

= 

m231and

D23

−

D13

= 

m221,implythat R

=

F1.Thiscompletestheproofof theresultreportedin[1] andgivenineqs. (10) and(11).

Twocommentsareinorder.First,thefunctionF

(θ

12

,

θ

13

,



m221

,



m2

Fig. 6. ThefunctionsF (bluesolidline), F (red˜ dashedline)andthedifferenceF− ˜F (greenline)versusA/m2

21 forNO(leftpanel)andIO(rightpanel)neutrinomass spectrum.Seetextforfurtherdetails.(Forinterpretationofthecoloursinthefigure(s),thereaderisreferredtothewebversionofthisarticle.)

fromeqs. (61) and(62) thatthefunctions F2 andF3 arepositive. ForA

=

0,wehavealsoF1D12D13D23D32

>

0.Onecanshowthat thisinequalityholdsalsoforA

=

0,whichleadstoF

>

0.This im-pliesthat therephasing invariants invacuumand inmatter, JCP and Jm,havethesamesign:

sgn



Jm



=

sgn



JCP



.

(67)

This result is valid both for neutrino mass spectra with normal ordering(m2₃₁

>

0)andwithinvertedordering(m2₃₁

<

0).

Second,thefunction F

(θ

12

,

θ

13

,



m₂₁2

,



m2₃₁

,

A

)

ineq. (10) has differentequivalentrepresentations.Thisshouldbe clearfromthe factthat JCP

(

Jm

)

=

Im



U(_μm₂)



_∗ U(_e2m)U_μ(m₃)



U(_e3m)



_∗



=

Im



U(_μm₃)



_∗ U(_e3m)U_μ(m₁)



U_e1(m)



_∗



=

Im



U(_μm₂)



U(_e2m)



_∗



U_μ(m₁)



_∗ U_e1(m)



= ... ,

(68)

andthederivationpresentedabove.Indeed,wecanusethesecond or the third form of JCP ( Jm) in eq. (68) to obtain the relation givenineq. (10). Thefunction F thus derived willdifferin form from,butwillbeequalto,thefunction F definedineqs. (11)–(15).

Itfollowsfromeqs. (10) and(19) that

Jm JCP

=

F

(θ

12

, θ

13

,



m221

,

m231

,

A

)

= ˜

F

=



m2 12



m223



m231



M₁₂2



M2₂₃



M2₃₁

,

(69)

i.e.,thatthefunctionF

(θ

12

,

θ

13

,



m212

,



m231

,

A

)

foundin[1] is an-otherrepresentationofthefunction F found

˜

in[2].Thefunctions

F and F have

˜

very differentforms.Nevertheless,aswe have ver-ified,they coincidenumerically.ThisisillustratedinFig. 6where weshowthefunctionsF (eq. (11)),F (eq. (

˜

20))andthedifference

(

F

− ˜

F

)

versus A

/

m2₂₁.WeusedtheanalyticexpressionsforM2_i,

i

=

1,2,3,in termsofm2

1, A andthe neutrinooscillation param-eters



m2₃₁,



m2₂₁,

θ

12 and

θ

13 derived in[12].Itshouldbe clear fromeq. (22) that,aswehavealreadydiscussed,inthe parametri-sation(7) employed in[1] themassparametersM2

i,i

=

1,2,3,do notdependon

θ

23and

δ.

InFig.6,theneutrinooscillation parame-tersonwhichthefunctionsF andF depend

˜

weresettotheirbest fit valuesfound in the globalanalysisof the neutrinooscillation datain[14] inthecasesofNOandIOneutrinomassspectra.

As issuggestedbyFig.6 andwe havecommentedearlier,our numericalresultsshowthatthefunction F ispositive.

The function F in eq. (10), aswe haveremarked earlier,does notdependon

θ

23and

δ.

Thisimpliesthattheratio

Jm

sin 2

θ

23sin

δ

=

F JCP

sin 2

θ

23sin

δ

=

1

8F cos

θ

13sin 2

θ

12sin 2

θ

13

,

(70)

doesnotdependon

θ

23and

δ.

Fromeqs. (11),(61),(62) and(50),usingUm_e1

=

Ue1D21D31

/

D1 wefindanewexpressionforthefunction F1 aswell:

F1

=

D2D3

D1

D21D31

.

(71)

4. Thecaseofantineutrinomixinginmatter

In the preceding Sections we have focused primarily on the mixingandoscillationsinmatterofflavourneutrinos.Inthis Sec-tion wewilldiscussbriefly thecaseofmixingandoscillationsin matterofflavourantineutrinos.

In ordinary matter (of, e.g., theEarth, the Sun) the mixingof antineutrinos in matter differs from the mixing of neutrinos in matter asa consequence of the fact that ordinary matter is not charge conjugation invariant: it contains protons, neutrons and electrons, butdoesnot containtheir antiparticles.Thiscauses CP andCPTviolatingeffectsinthe mixingandoscillationsof neutri-nosinmatter[6].Asaconsequence,theneutrinoandantineutrino mixingangles,aswellthemassesoftherespectiveneutrino mass-eigenstates, in matterdiffer. The expressions forthe antineutrino mixinganglesinmatter,

θ

i j,theneutrinomassesinthiscase, Mk, and the corresponding J -factor, Jm, can be obtainedfrom those corresponding toneutrinomixinginmatter,asiswell known,by replacingthepotential A with

(

−

A

).

Since the derivations ofthe resultsgiven ineqs. (69)–(71) do not depend on the sign of the matter term A, these results are validalsoformixingofantineutrinosinmatterandforoscillations ofantineutrinos

ν

l inmatterwithconstantdensity.Thus,wehave:

Jm

=

JCPF

(θ

12

, θ

13

,



m221

,



m231

,

A

) ,

(72)

F

(θ

12

, θ

13

,



m212

,



m231

,

A

)

=

F

(θ

12

, θ

13

,



m221

,



m231

,

−

A

) ,

Jm

=

Im



Um_e2

 

Um_μ3

 

Um_e3



∗



Um_μ2



_∗



=

1 8 cos

θ

m

13sin 2

θ

m12sin 2

θ

m23sin 2

θ

m13sin

δ

m

,

(74) whereUm_lj arethe elementsofthe antineutrinomixingmatrixin matter Um,

δ

m is theDiracphase presentin Um, and

θ

m_{i j} arethe antineutrinomixinganglesinmatter.Wealsohave:

sin 2

θ

m₂₃sin

δ

m

=

sin 2

θ

23sin

δ ,

(75)

F

=

U m e1Ume2Ume3 Ue1Ue2Ue3

=



m2₁₂



m2₂₃



m2₃₁



M2₁₂



M2₂₃



M2₃₁

,

(76)

with M2_{i j}

=

M2_i

−

M2_j. From the exactrelations (48) and(75) we get

sin 2

θ

₂₃m sin

δ

m

=

sin 2

θ

₂₃m sin

δ

m

=

sin 2

θ

23sin

δ ,

(77) whileeqs. (69) and(76) imply:

F

=

F



M

2

12



M223



M231



M2₁₂



M2₂₃



M2₃₁

.

(78)

Finally,asintheneutrinomixinginmattercase,theequalities

θ

m₂₃

= θ

23

,

0

< θ

23

, θ

m23

≤

π

/

2

,

(79)

δ

m

= δ ,

(80)

although not exact, represent excellent approximations for the rangesofvaluesof A

/

m2

21 relevant fortheT2K (T2HK),T2HKK, NO

ν

AandDUNEneutrinooscillationexperiments.

5. Summary

Inthe presentarticle we have analysed aspects of 3-neutrino mixinginmatter andofCPandTviolationin 3-flavourneutrino oscillations in vacuum andin matter withconstant density. The analyseshavebeenperformedintheparametrisationofthePMNS neutrinomixingmatrix UPMNS

≡

U specifiedineq. (7) and intro-ducedin[1].However, aswe haveshown, theresultsobtainedin ourstudyarevalid(insome caseswithtrivialmodifications)also inthestandardparametrisationofthePMNSmatrix(see,e.g.,[3]). Investigatingthecaseof3-neutrinomixinginmatterwith con-stant density we have derived first the relations

θ

₂₃m

= θ

23 and

δ

m

= δ

,

θ

23 (θ23m) and

δ

(δm) being respectively the atmospheric neutrinomixingangleandtheDiracCPviolationphaseinvacuum (inmatter)presentinthePMNSneutrinomixingmatrix. Perform-ing a detailed numerical analysis we have shown that although theseequalitiesarenotexact,theyrepresentexcellent approxima-tions for the ranges of values of A

/

m2₂₁

<

30 relevant for the T2K(T2HK),T2HKK,NO

ν

AandDUNEneutrinooscillation experi-ments,thedeviationsfromeachofthetworelationsnotexceeding respectively1.3

×

10−2and1.3

×

10−3(Figs.1and

2

)).Similar con-clusionisvalidforthecorrespondingparameters

θ

m₂₃and

δ

minthe caseofmixingofantineutrinos(Figs.3and

4

)).

Wehavederivednexttherelationsin 2θ₂₃msin

δ

m

=

sin 2θ23sin

δ,

andhave shownnumericallythatit isexact(Fig. 5). Therelation iswell knownintheliterature(see[23,13]).Wehavepresenteda newderivationofthisresult. Usingtheindicatedrelationandthe factthat thedeviations of

θ

₂₃m from

θ

23,

23

(

A

/

m221

),

andof

δ

m from

δ,

δ

(

A

/

m2

21

),

are small,

|

23

|,

|

δ

|



1,we havederived a

relationbetween

23and

δ

workinginleadingorderinthesetwo parameters (eq. (51)). It follows from this relation, in particular, that for

θ

23

=

π

/4 and

δ

=

3

π

/2,

π

/2,

the leading order matter correctionto

δ

vanishes,whilefor

δ

=

3

π

/2 (

π

/2)

and

θ

23

=

π

/4,

theleadingordermattercorrectionto

θ

23vanishes.

We have discussed further the relation between the rephas-ing invariants, associated with the Dirac phase in the neutrino mixing matrix, which determine the magnitude of CP and T vi-olating effects in 3-flavour neutrino oscillations in vacuum, JCP, and of the T violating effects in matter with constant density,

Jm_T

≡

Jm_{, obtained in [1]: J}m

₌

_J

CPF . F is a function whose explicit form in terms of the squared masses in vacuum and in matter of the mass-eigenstate neutrinos, of the solar and reac-tor neutrino mixing angles and of the neutrino matter poten-tial (eq. (11)) was given in [1]. The quoted relation between

Jm and JCP was reported in [1] without a proof. We have pre-sented a derivation of this relation. We have shown also that the function F

=

F

(θ

12

,

θ

13

,



m₂₁2

,



m2₃₁

,

A

)

i) is positive, F

>

0, which implies that Jm and JCP have the same sign, sgn(Jm

)

=

sgn(JCP

),

and that ii) it can have different forms. We have proven also that the function F as given in [1] is another rep-resentation ofthe so-called called“Naumov factor” (Fig. 6): F

=



m2₁₂



m2₂₃



m2₃₁

(

M2₁₂



M₂₃2



M2₃₁

)

−1_{, where}

_

_m2 i j

=

m 2 i

−

m 2 j,



M2_{i j}

=

M2_i

−

M2_j,mi andMi,i

=

1,2,3,beingthemassesofthe threemass-eigenstateneutrinosinvacuumandinmatter.

Finally,wehaveconsideredbrieflythecaseofantineutrino mix-inginmatterandhaveshownthatresultssimilartothosederived forthemixingofneutrinosinmatterarevalidalsointhiscase.

Theresultsofthepresentstudycontributetotheunderstanding oftheneutrinomixinginmatterandflavour neutrinooscillations inmatter withconstant density,widelyexplored intheliterature on the subject. Theycould be useful for the studies of neutrino oscillations inlong baseline neutrinooscillation experimentsT2K (T2HK),T2HKK,NO

ν

AandDUNE.

Acknowledgements

This work was supported in part by the INFN program on Theoretical Astroparticle Physics (TASP), by the European Union Horizon 2020 research and innovation programme under the MarieSklodowska-Curiegrants674896 and690575,by theWorld Premier International Research Center Initiative (WPI Initiative, MEXT),Japan(S.T.P.),aswellasbytheEuropean ResearchCouncil underERCgrant“NuMass”FP7-IDEAS-ERCERC-CG617143(Y.L.Z.).

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