Int J Theor Phys (2014) 53:727–733DOI 10.1007/s10773-013-1860-4

Neutrino Mass Hierarchy and CP Violation in LongBaseline Neutrino Experiment

Bipin Singh Koranga

Received: 5 August 2013 / Accepted: 1 October 2013 / Published online: 13 October 2013© Springer Science+Business Media New York 2013

Abstract We discuss the CP violation in long base line neutrino oscillation experiments.The direct measurement of CP violation is the difference of transitions probability betweenCP conjugate channels. The sign of Δ31 is not yet determined, we assume two mass hi-erarchy conditions, normal (Δ31 > 0) and inverted (Δ31 < 0). In this paper, we study theCP violation and neutrino mass hierarchy effect in vacuum and matter for long baselineBNL experiments. By an appropriate chose of experimental parameter, neutrino energy andtraveled distance. We find that, in matter normal mass hierarchy en-chanced maximum CPviolation over their invert mass hierarchy value by 12 %.

Keywords Neutrino mass hierarchy · CP Violation

1 Introduction

The origin of CP violation is still mystery in particle physics, recent advance in neutrinophysics observation mainly of astrophysical observation suggested the existence of tiny neu-trino mass. The experiments and observation have shown evidences for neutrino oscillation.The Solar neutrino deficit has long been observed [1–4], the atmospheric neutrino anomalyhas been found [5–7] and currently almost confirmed by KamLAND [8], and hence indi-cates that neutrino are massive and there is mixing in lepton sector. Since there is a mixingin lepton sector, this indicate to imagine that there occurs CP violation in lepton sector.Several physicist have considered whether we can see CP violation effect in lepton sectorthrough long baseline neutrino oscillation experiments. The neutrino oscillation probabili-ties, in general depends on six parameter two independent mass-square difference Δ21 andΔ31, there mixing angle Θ12,Θ23 and Θ13, and one CP violating phase δcp .

CP-violation effect arise as three (or more) generation [9, 10]. CP violation in neutrinooscillation is interesting because it relates directly to CP phase parameter in the mixingmatrix for n > 3 degenerate neutrinos. In this paper we consider the case of three light

B.S. Koranga (B)Department of Physics, Kirori Mal College (University of Delhi), Delhi 110007, Indiae-mail: bipiniitb@rediffmail.com

728 Int J Theor Phys (2014) 53:727–733

neutrinos. We write down the compact formula’s for the difference of transition probabilitybetween conjugate-channel.

ΔP = P (να → νβ) − P (ν̄α → ν̄β), (1)

where

(α,β) = (e,μ), (μ, τ), (τ, e).

CP violation in neutrino oscillation can in principal be observed in neutrino experimentsby looking at the difference of the transition probabilities between CP conjugate channel.Much progress has been made to works, determining the values of the three mixing angle.From measurement of the neutrino survival probability νμ → νe and νe → νe in the atmo-spheric flux, so that one mixing angle is near Π

4 and one is small [11] from the νe → νe

survival probability in the solar flux, so that the mixing angle is either large (LMA) or small(SMA) by solar solution [12]. Nothing is known about CP violating phase, we show that foran appropriate choice of neutrino oscillation parameter decrease CP asymmetry in matterfor BNL experiments. The purpose of this paper is to present the numerical study of CP vi-olation and neutrino mass hierarchy in long baseline BNL experiment. In Sect. 2, we reviewsome basic parameter of CP violation in vacuum, we also summarize the current oscillationparameter. In Sect. 3 CP violation in vacuum and in matter. In Sect. 4 we give the result ofyour numerical computing of maximum CP violation.

2 CP Violation Parameter and Neutrino Masses Difference

The solar neutrino deficit and atmospheric neutrino deficit explain by neutrino oscillation.In general the weak eigenstate is difference from the mass eigenstate as in the quark sector,this given rise to the so called neutrino oscillation phenomena. Strong evidence for neutrino-oscillation have been indicate the finite mass and mixing of neutrinos. The first evidence isthe observation of zenith-angle dependence of atmospheric neutrino defect [13] dependentof the atmospheric neutrino νμ → ντ transition with the mass difference and the mixing as

Δ31 = 1 − 2 × 10−3 eV2, sin2 2θ23 = 1.0 (2)

where Δij = m2i − m2

j and mi is the neutrino mass.The second evidence is the solar neutrino deficit [14]. Which is consistent with νe →

νμ/ντ transition. The SNO experiments [15] are consistent with the standard solar modeland Strong suggest the LMA solution

Δ21 = 7 × 10−5 eV2, sin2 2θ12 = 0.8. (3)

The main physical goals in future experiments are the determination of the unknown pa-rameter θ13, and upper bound sin2 2θ13 < 0.1 is obtained [16]. In particular, the observationof δcp is quite interesting from the point of view that δcp is related to the origin of the matterin the universe. The determination of δcp is the final goal in the neutrino oscillation exper-iments. We get analytical expression for ΔP and Acp using the usual form of the CKMmatrix parametrization

U =⎛⎝

c12c13 s12c13 s13e−iδcp

−s12c23 − c12s23s13eiδcp c12c23 − s12s23s13e

iδcp s23c13

s12s23 − c12c23s13eiδcp −c12s23 − s12s13e

iδcp c23s13

⎞⎠ , (4)

Int J Theor Phys (2014) 53:727–733 729

where c and s denoted the cosine and sine of the respective notation, thus ΔP in vacuumcan be written as

ΔP = P (να → νβ) − P (ν̄α → ν̄β) = 16Jcp(sinΔ21 sinΔ32 sinΔ31) (5)

here α and β denote different neutrino (or anti-neutrino) flavors.Where

Δij = 1.27

(Δ2mij

1 eV2

)(L

Km

)(1 GeV

E

)(6)

Δ2mij = (m2i − m2

j ), is the difference of ith and j th vacuum mass-square eigenvalues, E isthe neutrino energy, and L is the travels distance and the well known jariskog-invariance[19, 20] Jcp in the standard mixing parametrization is given by

jcp = 1

8sin2 2Θ12 sin2 2Θ13 sin2 Θ23 cosΘ13 sin δcp (7)

and the asymmetry parameter suggested by Cabibbo [14], as an alternative to measure CPviolation in the lepton sector

Acp = ΔP

P(να → νβ) + P (να → νβ). (8)

The difference of transition probability between CP conjugated channel Δp depend mix-ing angle Θ12,Θ23,Θ13, CP phase δcp and mass difference square Δ21,Δ31. The sign ofΔ31 is not yet determine. In this paper we check the effect of mass hierarchy in vacuum CPasymmetry and matter CP asymmetry in neutrino oscillation [15].

2.1 ΔP(α,β) = P (να → νβ) − P (ν̄α → ν̄β) in Vacuum

Probability difference ΔP(α,β) in term of two different mass-square eigenvalues differencein vacuum can be written as

ΔP v(α,β)

= 8jcp

(sin

(2.54Δ31l

E

)sin2

(1.27Δ21l

E

)− sin2

(1.27Δ31l

E

)sin

(2.54Δ21l

E

))(9)

where Δ21 and Δ31 is in eV2, the base line length L is in Km and neutrino energy E isin GeV.

2.2 ΔP = P (να → νβ) − P (ν̄α → ν̄β) in Matter

In the long base line experiments, the neutrino passing through earth’s crust which has con-stant density of about 3 gm/cc, in our discussion below we use the approximation of CPphase ΔP(α,β) is unaffected by matter effect. The important of matter effect on long base-line neutrino oscillation experiment has been recognized for many years. In this paper weused analytic formula of νμ → νe for three generation mixing, obtained with the assumptionof constant matter density [17, 18]. The oscillation of νμ → νe is the earth for 3-generationmixing is described approximately by given equation

P (νμ → νe) = sin2 2Θ23sin2 2Θ13

(A1 − 1)2sin2

((A1 − 1)Δ

)

± α sin δcp cosΘ13 sin 2Θ12 sin 2Θ13 sin 2Θ23

A1(1 − A1)

730 Int J Theor Phys (2014) 53:727–733

× sin(Δ) sin(A1Δ) sin((1 − A1)Δ

)

+ α cos δcp cosΘ13 sin 2Θ12 sin 2Θ13 sin 2Θ23

A1(1 − A1)

× sin(Δ) cos(A1Δ) sin((1 − A)1Δ

)

+ α2 cos2 Θ23 sin2 2Θ12 sin2(A1Δ)

A12(10)

In Eq. (10) “+” stands for neutrinos and “−” for anti-neutrino.Probability difference ΔP(α,β) in term of mass-square eigenvalues difference in matter

can be written as

Δpm(α,β) = sin2 2Θ23 sin2 2Θ13

(sin2((1 − A1)Δ)

A1(1 − A1)2− sin2((1 + A1)Δ)

A1(1 + A1)2

)

8αjcp sin(Δ) sin(A1Δ)

(sin((1 − A1)Δ)

A1(1 − A1)+ sin((1 + A1)Δ)

A1(1 + A1)

)

8α cot(δcp) sin(Δ) cos(A1Δ)

(sin((1 − A1)Δ)

A1(1 − A1)− sin((1 + A1)Δ)

A1(1 + A1)

)(11)

where α = Δ21Δ31

,Δ = Δ31L

4E, A1 = 2

√2GF neE

Δ31and matter term A is given by

A = 2√

2GF neE = 0.76 × ρ (gm/cc) × E (GeV) × 10−4 eV2, (12)

where GF is the Fermi coupling constant, ρ is the density of the involving matter profileand E is the neutrino energy. Probability difference ΔP m(α,β) in matter Eq. (10) can besimplified further and written as

ΔP m(α,β) = 16αjcp sin(Δ) sin(A1Δ)

(sin((1 − A1)Δ)

A1(1 − A1)

). (13)

To illustrate how mass hierarchy affects the CP asymmetry in vacuum and matter. Wehave chosen appropriate value for the neutrino mass square difference and mixing angle,which are allowed for the atmospheric neutrino and the large mixing angle solar solution. Inthis paper we will investigated the possible value of ΔP in vacuum and in matter allowed bypresent experimental data and effect of normal and inverted mass hierarchy in CP asymmetryat long base line BNL experiments. We are interested in relation between matter term A andΔ31 for which ΔP m in matter effective than ΔP v in vacuum. Here we consider two differentsets of condition.

2.3 Matter Term A < Δ31 in the Energy Range (1–10) GeV

At energy range (1–10) GeV, matter term A = 2√

2GF neE in Eq. (12), A < Δ31. At partic-ular energies E1 = αΔ31

2√

2GF ne, leads to following condition:

ΔP m(α,β) = ΔP v(α,β) (14)

We note that, in current neutrino oscillation parameter, we get above condition, whenmatter term A = 3 %Δ31.

Int J Theor Phys (2014) 53:727–733 731

2.4 Matter Term A > Δ31 in the Energy Range (E > 10) GeV

At energy range (>10) GeV, matter term A = 2√

2GF neE in Eq. (14), A > Δ31. At partic-ular energies E2 ≈ Δ31

2√

2GF ne, leads to following condition:

ΔP m(α,β) = 0 (15)

We note that, when matter term A ≈ Δ31 in this case CP asymmetry in matter indepen-dent of CP phase δcp . In matter, the first peaks in the CP violation asymmetry given byEq. (12) occur when sin(Δ) sin(A1Δ) sin((1 − A1)Δ) is maximized, the rough location ofthese peaks is given by

L

E≈ (2n + 1)Π

2.54Δ31(16)

For the firs peaks, a numerical; calculation produce an estimate more accurate thenEq. (11) is given by

L

E≈ 7Π

(2n + 1)2.54Δ31(17)

First peaks of CP asymmetry occur in matter, when

Δ = 1.27Δ31L

E1= 7Π

2, (18)

E1 = 0.5 GeV.Second peaks of CP asymmetry occur in matter, when

Δ = 1.27Δ31L

E2= 7Π

3, (19)

E2 = 1.2 GeV.Third peaks occur of CP asymmetry occur in matter, when

Δ = 1.27Δ31L

E3= 7Π

5, (20)

E3 = 1.2 GeV.When

Δ = 1.27Δ31L

E4= Π.

In matter, after three peaks the size of maximum CP asymmetry is effected by masshierarchy. In normal mass hierarchy case,

Δpm(α,β,+Δ31) = 16αjcp sin3(Δ), (21)

Δpm(α,β,−Δ31) = −16αjcp sin3(Δ). (22)

Maximum CP asymmetry difference for normal and invert mass hierarchy is given by

Δpm(α,β,+Δ31) − Δpm(α,β,−Δ31) = 32αjcp sin3(Δ). (23)

732 Int J Theor Phys (2014) 53:727–733

Fig. 1 ΔP probabilities difference vs E for Δ31 = +0.002 eV2,Δ21 = 0.00008 eV2Θ13 = 10◦, δcp = 90◦and L = 2540 km. The solid line is ΔP in vacuum and the dotted line is ΔP in matter

3 CP Violation in Long Baseline Experiments: A Numerical Analysis

Now we can calculate the maximum value of ΔP = P (νμ → νe) − P (ν̄μ → ν̄e) in vacuumand matter, which is the direct measurement of CP violation. The maximum values of ΔP invacuum and matter are presented for fixed value of L, E and different mass hierarchy con-dition. We show numerical result for cases of L = 2540 km. Here we used the CP effectiveenergy band of E = 1.0 to 3.0 GeV. In this section we present the result of our numericalanalysis using Eqs. (9) and (11).

We summarize the following observation from Figs. 1 and 2.

(1) Energy range (0.5–1.8) GeV, effect of normal mass hierarchy is almost same in vacuumand in matter. In this energy region normal and inverted mass hierarchy increase CPasymmetry in vacuum and matter.

(2) In vacuum, above energy E > 1.0 GeV normal mass hierarchy increase maximum CPasymmetry change by 14 % and inverted mass hierarchy increase by 13 %.

(3) In matter, above energy range E > 1.8 GeV sign of Δ31 play important role in CPasymmetry. Normal mass hierarchy increase maximum CP asymmetry by 12 % and in-vert mass hierarchy decrease CP asymmetry by 2 %. This is true for present experimentsvalues of oscillation parameter and maximum CP phase.

4 Conclusion

We have studied the direct measurement of CP violation originated by the phase of neutrinomixing matrix in the proposed BNL experiment. In conclusion, we have investigate in detailthe CP violation in long baseline and effect due to mass hierarchy in CP violation. We have

Int J Theor Phys (2014) 53:727–733 733

Fig. 2 ΔP probabilities difference vs E for Δ31 = −0.002 eV2,Δ21 = 0.00008 eV2Θ13 = 10◦, δcp = 90◦and L = 2540 km. The solid line is ΔP in vacuum and the dotted line is ΔP in matter

found, in matter by an appropriate choose of experimental parameter normal mass hierarchyincrease CP asymmetry over their invert mass hierarchy is about 12 %. In BNL experiment1–3 GeV energy region CP phase more effective than matter effect. In this energy regionmatter effect decrease CP asymmetry.

Acknowledgements The author would like to thank K.M.C-101, innovation project team for their com-ments and discussion on the manuscript.

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