xing, zhou

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Majorana CP-violating phases in neutrino-antineutrino oscillationsand other lepton-number-violating processes

Zhi-zhong Xing Ye-Ling Zhou

Institute of High Energy Physics, Chinese Academy of Sciences, P.O. Box 918, Beijing 100049, ChinaTheoretical Physics Center for Science Facilities, Chinese Academy of Sciences, Beijing 100049, China

Abstract

If the massive neutrinos are identied to be the Majorana particles via a convincing measurementof the neutrinoless double beta (0 ) decay, how to determine the Majorana CP-violating phases inthe 3 3 lepton avor mixing matrix U will become a desirable experimental question. The answerto this question is to explore all the possible lepton-number-violating (LNV) processes in which theMajorana phases really matter. In this paper we carry out a systematic study of CP violation inneutrino-antineutrino oscillations, whose CP-conserving parts involve six independent 0 -like massterms m and CP-violating parts are associated with nine independent Jarlskog-like parametersV

ij (for , = e,, and i, j = 1, 2, 3). With the help of current neutrino oscillation data, we

analyze the sensitivities of | m | and V ij to the three CP-violating phases of U , and illustrate

the salient features of six independent CP-violating asymmetries between and oscillations. As a by-product, the effects of the CP-violating phases on the LNV decays of doubly-and singly-charged Higgs bosons are reexamined by taking account of the unsuppressed value of 13 . Such CP-conserving LNV processes can be complementary to the possible measurements of neutrino-antineutrino oscillations in the distant future.

PACS number(s): 14.60.Pq, 13.10.+q, 25.30.PtKeywords: Majorana phases, neutrino-antineutrino oscillations, CP violation

E-mail: [email protected] E-mail: [email protected]

1

a r X i v : 1 3 0 5 . 5 7 1 8 v 2 [ h e p - p h ] 2 9 J u l 2 0 1 3


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1 Introduction

Neutrinos are the most elusive fermions in the standard electroweak model, partly because they areelectrically neutral and their masses are too small as compared with those charged leptons and quarks.The neutrality and smallness of neutrinos make it experimentally difficult to identify whether they arethe Dirac or Majorana particles, but most theorists believe that massive neutrinos should have the

Majorana nature (i.e., they are their own antiparticles [ 1]). To verify the Majorana nature of massiveneutrinos, the most feasible way up to our current experimental techniques is to detect the neutrinolessdouble beta (0 ) decay of some even-even nuclei [2]: A(Z, N ) A(Z + 2 , N 2) + 2 e , in whichthe lepton number is violated by two units. However, the 0 decay is a CP-conserving process andcannot directly be used to probe the Majorana CP-violating phases. Hence one has to consider otherpossible ways out of such a situation.

Given three massive neutrinos of the Majorana nature, the 3 3 Pontecorvo-Maki-Nakagawa-Sakata(PMNS) matrix U [3] can be parametrized in terms of three avor mixing angles ( 12 , 13 , 23 ) and threeCP-violating phases ( ,,) as follows:

U =c12 c13 s12 c13 s13 e i

s12 c23 c12 s13 s23 ei c12 c23 s12 s13 s23 ei c13 s23s12 s23 c12 s13 c23 ei c12 s23 s12 s13 c23 ei c13 c23

ei 0 00 ei 00 0 1

, (1)

where cij cosij and sij sin ij (for ij = 12 , 13, 23). Although is usually referred to as theDirac CP-violating phase which naturally appears in those lepton-number-conserving processes such asneutrino-neutrino and antineutrino-antineutrino oscillations, one should keep in mind that it is actuallya Majorana phase like or and can also show up in those lepton-number-violating (LNV) processessuch as the 0 decay and neutrino-antineutrino oscillations. This point will soon become clear. Sofar all the three neutrino mixing angles have been measured to a good degree of accuracy in a number of solar, atmospheric, reactor and accelerator neutrino oscillation experiments [4]. A determination of the

phase parameter via a measurement of the Jarlskog invariant J = c12 s12 c213 s13 c23 s23 sin [5] will be

one of the major goals of the next-generation long-baseline neutrino oscillation experiments. The mostchallenging task is to detect the Majorana phases and , which can only emerge in the LNV processes.As formulated by one of us in Ref. [ 6], it is in principle possible to determine all the three phasesfrom the CP-violating asymmetries A between and oscillations. Nevertheless, asystematic study of this problem has been lacking.

The present work aims to go beyond Ref. [ 6] by carrying out a systematic analysis of the MajoranaCP-violating phases in both neutrino-antineutrino oscillations and LNV decays of doubly- and singly-charged Higgs bosons based on the type-II seesaw mechanism [7] 1 , in order to reveal their distinctproperties which might be more or less associated with the observed matter-antimatter asymmetry of

the Universe [ 9]. Our study is different from the previous ones at least in the following aspects:

All the 0 -like mass terms m and the Jarlskog-like parameters V ij (for , = e, , and

i, j = 1 , 2, 3), which measure the CP-conserving and CP-violating properties of Majorana neutrinosrespectively, are analyzed in detail.

The sensitivities of all the CP-violating asymmetries A to the phase parameters and the neutrinomass spectrum are discussed in a systematic way, and the pseudo-Dirac case with vanishing and is also explored to illustrate why is of the Majorana nature.

1 As the 0 decay has been extensively discussed in the literature [8], here we shall not pay particular attention to it.

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Figure 1: The Feynman diagram for oscillations, where stands for the chirality ip in theneutrino propagator which is proportional to the mass m i of the Majorana neutrino i = i .

The CP-conserving LNV decays of H and H bosons are reexamined by taking account of theunsuppressed value of 13 reported by the Daya Bay [ 10] and RENO [11] Collaborations, and thedependence of their branching ratios on , and is investigated.

Such a comprehensive analysis of the Majorana phases in CP-violating and CP-conserving LNV processesshould be useful to illustrate how important they are in both lepton avor mixing and CP violation andhow difficult they are to be measured in reality.

The remaining parts of this paper are organized as follows. In section 2 we briey review the salientfeatures of three-avor neutrino-antineutrino oscillations, including a concise discussion about the CP-and T-violating asymmetries. Section 3 is devoted to a detailed analysis of six independent 0 -likemass terms m and nine independent Jalskog-like parameters V

ij (for , = e, , and i, j = 1 , 2, 3),

which appear in the probabilities of oscillations and their CP- or T-conjugate processes. Acomparison between V ij and J is made by switching off the Majorana phases and . As a by-product,the effects of three CP-violating phases on the LNV decays of doubly- and singly-charged Higgs bosonsare also reexamined by taking account of the unsuppressed value of 13 . In section 4 we carry out asystematic study of the sensitivities of six possible CP-violating asymmetries A to the three phaseparameters, the absolute scale and hierarchies of three neutrino masses, and the ratio of the neutrino

beam energy E to the baseline length L. Our numerical results illustrate the distinct roles of , and or their combinations in neutrino-antineutrino oscillations. Section 5 is devoted to a summary of thiswork with some main conclusions.

2 Salient features of neutrino-antineutrino oscillations

Let us consider oscillations (for , = e, , ), as schematically illustrated in Figure 1, wherethe production of and the detection of are both governed by the standard weak charged-currentinteractions. The amplitudes of transitions and their CP-conjugate processes can bewritten as [6, 12]

A( ) =i

U i U

im iE

exp im 2i2E

L K ,

A( ) =i

U i U im iE

exp im 2i2E

L K , (2)

where m i denotes the mass of i , E is the neutrino (or antineutrino) beam energy, L stands for the base-line length, K and K are the kinematical factors independent of the index i (and satisfying |K | = |K |).The helicity suppression in the transition between i and i is characterized by mi /E . The neutrino-antineutrino oscillation probabilities P ( ) |A( )|2 and P ( ) |A( )|2

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turn out to be [6]

P ( ) = |K |2E 2

m 2

4i


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where

| m |2

i

m 2i |U i |2 , (10)

which is actually the ( , ) element of M M . In particular, m e is just the effective mass term

appearing in the rate of the tritium beta decay 31 H 32 He + e + e . In comparison with Eq. (9), theso-called zero-distance effect of neutrino-antineutrino oscillations at L = 0 is given by

P ( ) = P ( ) = |K |2E 2

m 2

. (11)

Because of m i E , the effects in both Eqs. (9) and (11) are extremely suppressed.Thanks to CPT invariance, it is easy to check that P ( ) = P ( ) and P ( ) =

P ( ) hold. Hence the T-violating asymmetry between and oscillations must beexactly equal to the CP-violating asymmetry between and oscillations. To eliminatethe |K |2 /E 2 and |K |2 /E 2 factors, we dene the CP-violating asymmetry between and oscillations as the ratio of the difference P ( )P ( ) to the sum P ( )+ P ( ),denoted by

A [6]. Therefore,

A =2

i


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corresponding to m 221 7.5 10 5 eV2 and | m 231 | | m 232 | 2.4 10 3 eV2 , respectively. In thiscase, however, the sizes of the neutrino (or antineutrino) source and the detector must be much smallerthan the ones characterized by Losc21 and (or) Losc31 Losc32 . Note that the result in Eq. (11) is essentiallyequivalent to the Losc ji L case. If Losc ji L, instead, the m 2 ji -dependent oscillation terms will beaveraged out and then the probabilities will be simplied to

P ( ) = P ( ) = |K |2

E 2i

m 2i Cii . (14)

This CP-conserving result can be compared with the ones in Eqs. (9) and (11).

3 Properties and proles of V ij and m As shown in section 2, neutrino-antineutrino oscillations are closely associated with the effective massterms m and the CP-violating quantities V ij . The former may also appear in some other LNVprocesses in which CP and T symmetries are conserved. Let us explore the analytical properties andnumerical proles of

V ij

and

m

in some detail in this section.

3.1 The Jarlskog-like parameters V ijIt is well known that the strength of CP and T violation in normal neutrino-neutrino and antineutrino-antineutrino oscillations is measured by a single rephasing-invariant quantity, the so-called Jarlskogparameter J [5], dened through

Im U i U j U

j U

i = J

k

ijk , (15)

where U is the PMNS matrix. In terms of the standard parametrization of U given in Eq. (1), we have

J = c12 s12 c213 s13 c23 s23 sin . (16)

Therefore, a measurement of the CP-violating asymmetry between P ( ) and P ( ) or theT-violating asymmetry between P ( ) and P ( ) can only probe the Dirac phase [14].In contrast, the other two phases of U (i.e., and ) may contribute to the Jarlskog-like quantities V

ij

dened in Eq. (5), and thus they can in principle be measured in neutrino-antineutrino oscillations.By denition, the Jarlskog-like parameters V ij satisfy the relations

V ij = V ij = V ji = V ji , (17)

and V ii = 0; but V ij = 0 for i = j . With the help of Eq. (17), one may express V ij in terms of threedifferent V ij as follows:

V ije = 12 V

ij V ijee V ij ,

V ije = 12 V

ij V ijee V ij ,

V ij = 12 V

ijee V ij V ij . (18)

This result implies that only nine V ij are independent.

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To see the explicit dependence of each V ij on the CP-violating phases, let us calculate V

ij in the

standard parametrization of U given by Eq. (1). We obtain

V 12ee = c

212 s

212 c

413 sin 2 ( ) ,

V 13ee = c

212 c

213 s

213 sin 2 ( + ) ,

V 23ee = s

212 c

213 s

213 sin 2 ( + ) ; (19)

and

V 12 = c

212 s

212 c

423 4s

213 c

223 s

223 + s

413 s

423 sin2( )

+2 c12 s12 s13 c23 s23 c223 s

213 s

223 c

212 sin (2 2 + ) s

212 sin (2 2 )

+ s213 c223 s

223 c

412 sin 2 ( + ) + s

412 sin 2 ( ) ,

V 13 = c

213 s

223 s

212 c

223 sin2 + 2 c12 s12 s13 c23 s23 sin ( + 2 ) + c

212 s

213 s

223 sin 2 ( + ) ,

V 23 = c

213 s

223 c

212 c

223 sin2 2c12 s12 s13 c23 s23 sin ( + 2 ) + s

212 s

213 s

223 sin 2 ( + ) ; (20)

and

V 12 = c

212 s

212 s

423 4s

213 c

223 s

223 + s

413 c

423 sin2( )

2c12 s12 s13 c23 s23 s223 s

213 c

223 c

212 sin (2 2 + ) s

212 sin (2 2 )

+ s213 c223 s

223 c

412 sin 2 ( + ) + s

412 sin 2 ( ) ,

V 13 = c

213 c

223 s

212 s

223 sin 2 2c12 s12 s13 c23 s23 sin ( + 2 ) + c

212 s

213 c

223 sin 2 ( + ) ,

V 23 = c

213 c

223 c

212 s

223 sin2 + 2 c12 s12 s13 c23 s23 sin ( + 2 ) + s

212 s

213 c

223 sin 2 ( + ) . (21)

Taking account of Eq. (18), we can immediately write out the explicit expressions of V ij (for = )with the help of Eqs. (19)(21):

V 12

e =

c212 s

212 c

213 c

223

s213 s

223 sin2(

)

c12 s12 c213 s13 c23 s23 c

212 sin (2 2 + ) s

212 sin (2 2 ) ,

V 13e = c12 c

213 s13 s23 [s12 c23 sin ( + 2 ) c12 s13 s23 sin 2 ( + )] ,

V 23e = + s12 c

213 s13 s23 [c12 c23 sin ( + 2 ) s12 s13 s23 sin 2 ( + )] ; (22)

and

V 12e = c

212 s

212 c

213 c

223 s

213 s

223 sin2( )

+ c12 s12 c213 s13 c23 s23 c

212 sin (2 2 + ) s

212 sin (2 2 ) ,

V 13e = + c12 c

213 s13 c23 [s12 s23 sin ( + 2 ) c12 s13 c23 sin 2 ( + )] ,

V 23e = s12 c

213 s13 c23 [c12 s23 sin ( + 2 ) + s12 s13 c23 sin 2 ( + )] ; (23)

and

V 12 = c

212 s

212 c

423 s

213 1 + s

213

2c223 s

223 + s

213 s

423 sin2( )

c12 s12 s13 c23 s23 1 + s213 c

223 s

223 c

212 sin (2 2 + ) s

212 sin (2 2 )

s213 c

223 s

223 c

412 sin 2 ( + ) + s

412 sin 2 ( ) ,

V 13 = c

213 c23 s23 s

212 c23 s23 sin2 + c12 s12 s13 c

223 s

223 sin( + 2 ) + c

212 s

213 c23 s23 sin 2 ( + ) ,

V 23 = c

213 c23 s23 c

212 c23 s23 sin 2 c12 s12 s13 c

223 s

223 sin( + 2 ) + s

212 s

213 c23 s23 sin 2 ( + ) . (24)

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Table 1: The simplied expressions of the Jarlskog-like parameters V ij and their relations with the

Jarlskog parameter J in the = = 0 limit. Their typical numerical results are obtained by inputting12 33.4 , 13 8.66 and 23 40.0 [15] in the = 45 and = 90 cases.

Jarlskog-like parameter = 45 = 90

V 12ee = 0 0 0

V 13ee = c212 c213 s213 sin 2 +0 .016 0V 23ee = s212 c213 s213 sin 2 +0 .0067 0V 12 = 2J c223 s213 s223 + V 13ee V 23ee c223 s223 /c 213 +0 .030 +0 .039V 13 = 2J s223 + V 13ee s423 +0 .022 +0 .028V 23 = 2J s223 + V 23ee s423 0.018 0.028V 12 = 2J s213 c223 s223 + V 13ee V 23ee c223 s223 /c 213 0.017 0.027V 13 = 2J c223 + V 13ee c423 0.022 0.039V 23 = 2J c223 + V 23ee c423 +0 .030 +0 .039V 12e = J 0.024 0.033V

13e = J + V

13ee s

223 0.030 0.033V 23e = J V 13ee s223 +0 .021 +0 .033

V 12e = J +0 .024 +0 .033V 13e = J V 13ee c223 +0 .014 +0 .033V 23e = J V 23ee c223 0.028 0.033V 12 = J 1 + s213 s223 c223 V 13ee V 23ee c223 s223 /c 213 0.0064 0.0060V 13 = J c223 s223 + V 13ee c223 s223 +0 .0078 +0 .0058V 23 = J s223 c223 + V 23ee c223 s223 0.0025 0.0058

Similar expressions for Cij have been listed in Appendix A. These results clearly tell us how the CP-violating quantities V

ij depend on the CP-violating phases , and : (a) each V 12 is a function of

and (the only exception is V 12ee , which only involves ); (b) each V 13 is a function of and ; and (c) each V 23 is a function of and . The following extreme cases are particularly interesting.

In the = 0 (or ) limit, J = 0 holds, but all the V ij are in general nonvanishing. In this specialcase there will be no CP or T violation in normal neutrino-neutrino and antineutrino-antineutrinooscillations, but large CP or T violation in neutrino-antineutrino oscillations is possible.

In the 13 = 0 limit, J = 0 holds, so does

V 13ee = V

23ee = V

13e = V

23e = V

13e = V

23e = 0 , (25)

simply because all of them involve the U e3 = s13 ei element. Those nonvanishing Jarlskog-like

parameters depend on either or , or their difference . However, such an extreme case isnot favored by the recent reactor antineutrino oscillation data (i.e., 13 9 [10, 11]).

In the = = 0 limit, which looks like a pseudo-Dirac case with a single CP-violating parameter , we obtain V 12ee = 0, V 13ee = c212 c213 s213 sin 2 and V 23ee = s212 c213 s213 sin 2 . The other fteen V ijcan all be given in terms of J , V 13ee and V 23ee , as listed in Table 1. We see V 12e = V 12e = J ,

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V 12e e

0 . 2

0 . 1

0 . 0

0 . 1

0 . 2

V 12

0 . 2

0 . 1

0 . 0

0 . 1

0 . 2

V 12

0 . 2

0 . 1

0 . 0

0 . 1

0 . 2

0 45 90 13 5 18 0

V 13e e

V 13

V 13

0 45 90 13 5 180

V 23e e

V 23

V 23

0 45 90 135 18 0

Figure 2: The Jarlskog-like parameters V ij changing with the CP-violating phases. The green solid, reddashed, blue dotted and black dashed-dotted lines correspond to = 0 , 45 , 90 and 180 , respectively.

The typical inputs are 12 33.4 , 13 8.66 and 23 40.0 [15].and the other nonvanishing V ij may also receive the higher-order contributions proportionalto s213 sin 2 (i.e., the V 13ee and V 23ee terms). In this case the Jarlskog parameter J governs CPand T violation in both normal neutrino-neutrino (or antineutrino-antineutrino) oscillations andneutrino-antineutrino oscillations.

Of course, it is possible to relate V ij to J in some other special cases. For example, = = leadsto V 12e = V 13e = V 23e = V 12e = V 13e = V 23e = J .

We proceed to illustrate the numerical dependence of V ij on , and by taking 12 33.4 ,

13 8.66

and 23 40.0

as the typical inputs [ 15]. As the Dirac phase is expected to bedetermined earlier than the Majorana phases and , one may x the value of (for example, = 0 ,45 , 90 or 180 ) to show how V

ij can change with , or . Our numerical results of V

ij for the

= and = cases are shown in Figures 2 and 3, respectively. In addition, the magnitude of each

V ij in the = = 0 limit is illustrated in Table 1. Some comments are in order. It is amazing that V 12ee , V 23 , V 23 and V 23 can maximally reach about 20% in magnitude. In

comparison, J 1/ 6 3 9.6% constrains the strength of CP and T violation in normalneutrino-neutrino oscillations [ 16]. The reason for possible largeness of the above four Jarlskog-like parameters is simply that their leading terms are only slightly suppressed by s 212 or s 223 .

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V 12e

0 . 2

0 . 1

0 . 0

0 . 1

0 . 2

V 12e

0 . 2

0 . 1

0 . 0

0 . 1

0 . 2

V 12

0 . 2

0 . 1

0 . 0

0 . 1

0 . 2

0 45 90 13 5 18 0

V 13e

V 13e

V 13

0 45 90 13 5 180

V 23e

V 23e

V 23

0 45 90 135 18 0

Figure 3: The Jarlskog-like parameters V ij (for = ) changing with the CP-violating phases. Thegreen solid, red dashed, blue dotted and black dashed-dotted lines correspond to = 0 , 45 , 90 and

180 , respectively. The typical inputs are 12 33.4 , 13 8.66 and 23 40.0 [15].

The magnitudes of V 13ee and V 23ee are strongly suppressed, because both of them are proportional tos213 2.3%. We see that the magnitudes of V 13e , V 23e , V 13e and V 23e are modest, since their leadingterms are comparable with J . In other words, they are essentially constrained to be 10%.

V 12ee has nothing to do with the Dirac phase , as one can see in Eq. (19). The dependence of V 12 , V 13 and V 23 on is very weak, because this dependence is suppressed either by the factors13 c223 s223 2.6% or by the factor s213 2.3% as shown in Eq. (24).

The pseudo-Dirac case illustrated in Table 1 is interesting in the sense that appreciable CP- andT-violating effects are expected to show up in neutrino-antineutrino oscillations even the Majoranaphases and vanish. Namely, the Majorana neutrinos with only the Dirac CP-violating phasebehave very differently from the Dirac neutrinos 3 .

Therefore, it is in principle possible to determine all the three CP-violating phases in neutrino-antineutrinooscillations, in which the strength of CP and T violation is governed by V ij , whose maximal magnitudescould be larger than that of J by a factor of two or so. We shall come back to this point in section 4to analyze the CP-violating asymmetries between and oscillations.

3 This point can also be seen by examining their distinct renormalization-group running effects [ 17].

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3.2 The effective mass terms m The effective mass terms m dened in Eq. (4) are important to understand the origin of neutrinomasses, since they are simply the ( , ) elements of the symmetric Majorana neutrino mass matrixM in the basis where the avor eigenstates of three charged leptons are identied with their masseigenstates. Namely,

M =m ee m e m e m e m m m e m m

, (26)

where m = m has been taken into account. In the standard parametrization of U , we havem ee = m1 c

212 c

213 e

2i + m 2 s212 c

213 e

2i + m 3 s213 e

2i ,

m = m1 s12 c23 + c12 s13 s23 ei

2

e2i + m 2 c12 c23 s12 s13 s23 ei

2

e2i + m 3 c213 s

223 ,

m = m1 s12 s23 c12 s13 c23 ei

2

e2i + m 2 c12 s23 + s12 s13 c23 ei

2

e2i + m 3 c213 c

223 ,

m e = m 1 c12 c13 s12 c23 + c12 s13 s23 ei

e2i

+ m 2 s12 c13 c23 c12 s12 s13 s23 ei

e2i

+ m 3 c13 s13 s23 ei ,

m e = + m 1 c12 c13 s12 s23 c12 s13 c23 ei e2i m 2 s12 c13 c12 s23 + s12 s13 c23 e

i e2i

+ m 3 c13 s13 c23 ei ,

m = m 1 s12 s23 c12 s13 c23 ei c23 s12 + c12 s13 s23 e

i e2i

m 2 c12 s23 + s12 s13 c23 ei c12 c23 s12 s13 s23 e

i e2i + m 3 c213 c23 s23 . (27)

We see that a measurement of the three CP-violating phases is absolutely necessary in order to fullyreconstruct the neutrino mass matrix M . Without the information on and , it would be impossibleto model-independently look into the structure of M via a bottom-up approach. On the other hand, apredictive model of lepton avors should be able to specify the texture of M via a top-down approach,such that its predictions can be experimentally tested.

Note that | m |2 in Eq. (10) can be related to m as follows:

m 2

= | m |2 =

i

m 2i |U i |2 , (28)

It is obvious that all the | m |2 do not contain any information about the Majorana phases and ,but they may depend on the Dirac phase . Furthermore, we have

| m |2

=i

m2i = 3 m

21 + m

221 + m

231 = 3m

23 m

221 2 m

232 , (29)

where m 2ij m2i m 2 j (for i, j = 1 , 2, 3). A global analysis of current neutrino oscillation data hasgiven m 221 7.50 10 5 eV2 and m 231 2.473 10 3 eV2 (normal neutrino mass hierarchy) or m 232 2.427 10 3 eV2 (inverted neutrino mass hierarchy) [15]. Therefore,

Normal hierarchy :i

m 2i m221 + m

231 2.55 10

3 eV2 ,

Inverted hierarchy :i

m 2i m221 2 m

232 4.78 10

3 eV2 , (30)

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| m e e | ( eV)

IH

NH

10 4

10 3

10 2

10 1

| m | ( eV)

10 4

10 3

10 2

10 1

| m | ( eV)

m 1 or m 3 ( eV)

10 4

10 3

10 2

10 1

10 4

10 3

10 2

10 1

| m e | ( eV)

| m e | ( eV)

| m | ( eV)

10 4

10 3

10 2

10 1

Figure 4: The proles of | m | versus the lightest neutrino mass m1 (normal hierarchy or NH: redregion) or m 3 (inverted hierarchy or IH: green region).

where the lower bounds correspond to m1 = 0 (normal hierarchy) and m3 = 0 (inverted hierarchy),respectively. On the other hand, the sum of the three neutrino masses can also be written as

i

m i = m1 +

m 21 + m 221 +

m 21 + m 231

= m3 + m 23 m 232 + m 23 m 232 m 221 . (31)This sum has well been constrained thanks to the recent WMAP [18] and PLANCK [19] data, and itsupper bound is about 0 .23 eV at the 95% condence level [19]. One may then obtain the allowed rangeof the lightest neutrino mass by using the above inputs: 0 m 1 0.071 eV in the normal hierarchy; or0 m 3 0.065 eV in the inverted hierarchy.

Figure 4 illustrate the proles of six | m |. Our inputs are 12 33.4 , 13 8.66 and 23 40.0 ; m 221 7.5010 5 eV2 and m 231 2.47310 3 eV2 (normal hierarchy) or m 232 2.42710 3 eV2

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(inverted hierarchy) [ 15]. As for the three unknown phase parameters, we allow the Dirac phase torandomly vary between 0 and 360 , and allow the Majorana phases and to randomly vary between0 and 180 . We plot the results of | m | versus the lightest neutrino mass in Figure 4 by allowingthe latter to vary from 10 4 eV to 10 1 eV, where the upper bound is set by taking account of therecent PLANCK data [19]. To understand our numerical results, we have also made some analyticalapproximations for

m in Appendix B. Some discussions are in order.

Given the normal neutrino mass hierarchy, most of the random points of | m |, | m |, | m |are located in the region of 10 2 eV to 10 1 eV. This observation is also true for all the | m |in the inverted hierarchy. Such results are compatible with the analytical approximations made inAppendix B. The point is that the relevant | m | are dominated by | m 231 | | m 232 | 0.05eV when the lightest neutrino mass is sufficiently small, and all the | m | approach m 1 m 2 m 3 > 0.05 eV for a nearly degenerate neutrino mass spectrum.

The random points of | m ee |, | m e | and | m e | in the normal hierarchy are most likely to lie inthe region of 10 3 eV to 10 2 eV, especially when m21 m 231 . Their magnitudes are in generalsmaller than those in the inverted hierarchy. The reason is simply that

|m ee

| m 221 s212

2.6 10 3 eV and | m e | |m e | m231 s13 7.5 10 3 eV hold in the normal hierarchy,

while in the inverted hierarchy the dominant masses m1 m2 m 232 do not undergo thiss13 suppression (see Appendix B). In the limit where the lightest neutrino mass approaches zero or much smaller than m 221 , theallowed region of | m | in the normal hierarchy is narrower than that in the inverted hierarchy,

as shown in Figure 4, where the only exception is | m ee |. The reason can be seen from Eqs. (49)(52) in Appendix B: the dominant term of each | m | (for = ee) is proportional to m 231and its uncertainty is associated with m 221 in the normal hierarchy, while the uncertainty of the same effective mass term in the inverted hierarchy does not undergo this suppression. Becausethe two terms of m ee in Eq. (49) are almost comparable in magnitude, its magnitude involves arelatively large uncertainty in the normal hierarchy as in the inverted hierarchy.

In the m1 m2 m3 limit, which is guaranteed if the lightest neutrino mass is larger or muchlarger than | m 231 | | m 232 | 0.05 eV, the | m | in both normal and inverted hierarchiesshould have the same bounds. This is because the m i can be factored out from the expression of each | m |, making the latter insensitive to the ordering of the three masses. Such a feature hasessentially been reected in Figure 4 (see the limit of m1 0.1 eV or m3 0.1 eV), and it willbecome more obvious if m 1 (or m 3 ) runs to much larger values, such as 0 .2 eV or even 0.5 eV. Seealso Appendix B for some relevant analytical approximations in this case.

For some values of the lightest neutrino mass, | m | = 0 is always allowed, as shown in Figure 4,either in the normal hierarchy (e.g., | m ee | = 0 [20]) or in the inverted hierarchy (e.g., | m | =0), or in both of them (e.g., | m e | = 0). This kind of texture zeros implies that signicantcancellations can happen in | m | due to the unknown CP-violating phases [20, 21]. For instance, = may lead to | m e | |m e | 0 when the three neutrino masses are nearly degenerate,as one can see from Eq. (53) in Appendix B. Fortunately, it is impossible for all the | m | tobe simultaneously vanishing or highly suppressed, no matter what values m1 or m3 may take.Unfortunately, current experimental techniques only allow us to constrain | m ee | via a carefulmeasurement of the 0 decay [8].

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Table 2: The branching ratios of the LNV H + + decay modes in four different neutrino masshierarchies, where = 90 has been assumed.

Normal hierarchy m1 = 0 m1 = 0 .1 eV

B (H + e+ ) 0.03 0.31B (H + + ) 0.41 0.34B (H + + ) 0.56 0.35Inverted hierarchy m3 = 0 m3 = 0 .1 eV

B (H + e+ ) 0.21 0.32B (H + + ) 0.30 0.33B (H + + ) 0.49 0.35

Taking the upper bound of the sum of three neutrino masses as set by the recent PLANCK data (i.e.,m 1 + m 2 + m 3 < 0.23 eV at the 95% condence level [ 19]), we may obtain the upper bounds on all the

| m | from our numerical calculations:| m ee | 0.072 eV , | m | 0.077 eV , | m | 0.080 eV ,| m e | 0.060 eV , | m e | 0.055 eV , | m | 0.078 eV ,

for the normal neutrino mass hierarchy; and

| m ee | 0.082 eV , | m | 0.075 eV , | m | 0.072 eV ,| m e | 0.065 eV , | m e | 0.058 eV , | m | 0.072 eV ,

for the inverted neutrino mass hierarchy.

3.3 H ++ ++ and H

+ + decaysThere exist a number of viable mechanisms which can explain why the neutrino masses are naturallytiny [22]. Among them, the type-II seesaw mechanism [ 7] is of particular interest because it can keepthe unitarity of the PMNS matrix U unviolated and lead to very rich collider phenomenology [23]. Thelatter includes the LNV decay modes H ++ +

+ and H

+ + . Their branching ratios are

B (H ++

+

+ )

(H ++ +

+ )

(H ++ +

+ )

= 21 +

m 2

i

m 2i, (32)

and

B (H +

+ )

(H + + )

(H + + )

= | m |2

i

m 2i, (33)

respectively, where the Greek subscripts run over e, and . Taking account of Eq. (28), we see that

B (H + + ) only depends on the Dirac phase , while B (H ++ ++ ) is sensitive to all the three

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0 . 0

0 . 2

0 . 4

0 . 6

0 . 8

1 . 0NH , m 1 = 0 eV

e e

e

e

NH , m 1 = 0 . 1 e V

I H, m 3 = 0 e V

0 45 9 0 135 180 0 . 0

0 . 2

0 . 4

0 . 6

0 . 8

1 . 0

B r a n c h i n g R a t i o s

I H, m 3 = 0 . 1 e V

0 45 90 13 5 180

Figure 5: The branching ratios of the LNV H ++ ++ decays as functions of the Majorana phase ,

where = 0 and = 90 are taken. Four typical cases of the neutrino mass spectrum are considered:the normal hierarchy (NH) with m1 = 0 or 0.1 eV; and the inverted hierarchy with m3 = 0 or 0.1 eV.

CP-violating phases. These interesting LNV decay modes deserve a reexamination because the previousworks [23, 24] were more or less subject to the assumption of vanishing or very small 13 , making therole of unimportant. In view of the experimental fact that 13 is not that small [10, 11], we updatethe numerical analysis of B (H + + ) and B (H ++ +

+ ) by taking the same inputs as above. Our

results are presented in Table 2 and Figures 57, respectively.We rst look at the branching ratios B (H + + ), whose magnitudes are governed by

| m |2 =

i

m 2i |U i |2 = m21 1 |U 3 |

2 + m 23 |U 3 |2 + m 221 |U 2 |

2 , (34)

in which only the U 2 elements (for = e, , ) contain , as shown in Eq. (1). Hence the contributions

of to | m | and B (H + + ) are suppressed not only by the smallness of 13 but also by thesmallness of m 221 . In particular, B (H + e+ ) is exactly independent of . These LNV decay modesare actually not useful to probe the Dirac phase . We use the typical inputs of three neutrino mixingangles and two neutrino mass-squared differences given above to calculate B (H + + ), and list thenumerical results in Table 2, where = 90 has been assumed. When varying from 0 to 360 , wend that the -induced uncertainties of all the branching ratios are lower than 1%.

Now let us turn to the branching ratios of the LNV H ++ ++ decays. We take ( , ) = (0

, 90 ),(45 , 0 ) and (45 , 90 ) to show how B (H ++ +

+ ) changes with in Figures 57, respectively.

Both the normal hierarchy with m 1 = 0 (or 0 .1 eV) and the inverted hierarchy with m 3 = 0 (or 0 .1 eV)

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0 . 0

0 . 2

0 . 4

0 . 6

0 . 8

1 . 0NH , m 1 = 0 eV

e e

e

e

NH , m 1 = 0 . 1 e V

I H, m 3 = 0 e V

0 45 9 0 135 180 0 . 0

0 . 2

0 . 4

0 . 6

0 . 8

1 . 0


I H, m 3 = 0 . 1 e V

0 45 90 13 5 180



are considered in each of the gures. Some discussions are in order.

The sum of the six independent branching ratios is equal to one, as guaranteed by Eq. (32) andthe unitarity of the PMNS matrix U . This point can be clearly seen in each gure, which is exactlysaturated by six different branching ratios.

In the normal neutrino mass hierarchy with m 1 = 0, the magnitude of m ee is highly suppressed,and thus B (H ++ e+ e+ ) 0. In this special case B (H ++ e+ + ) and B (H ++ e+ + ) arealso very small. In the inverted neutrino mass hierarchy with m3 = 0, the H ++ + + channelis strongly suppressed.

The Majorana phases and play an important role in all the six LNV decay modes. They maysignicantly affect the branching ratio of each process, making themselves easier to be detected.Given some specic values of and , each B (H ++ +

+ ) changes as a simple trigonometric

function of . When changes from one given value to another, the prole of the branching ratioof each decay mode will more or less shift and deform.

In some cases, the three CP-violating phases may give rise to large cancellations in m , makingsome of the LNV decay modes signicantly suppressed. In the ( , ) = (0 , 90 ) case, for example,the H ++ e+ e+ , H ++ + + and H ++ + + channels are somewhat suppressed when thelightest neutrino mass is about 0 .1 eV. It is therefore difficult to detect them.

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0 . 0

0 . 2

0 . 4

0 . 6

0 . 8

1 . 0NH , m 1 = 0 eV

e e

e

e

NH , m 1 = 0 . 1 e V

I H, m 3 = 0 e V

0 45 9 0 135 180 0 . 0

0 . 2

0 . 4

0 . 6

0 . 8

1 . 0


I H, m 3 = 0 . 1 e V

0 45 90 13 5 180



The Dirac phase , whose effect is always suppressed by the smallness of 13 , has relativelysmall inuence on the branching ratios of the LNV H ++ ++ decays. A comparison between

Figures 6 and 7 indicates that the relevant numerical results do not change much when changesfrom 0 to 90 . But the interplay of and (or ) is sometimes important.

The branching ratios in the normal hierarchy with m1 = 0 .1 eV and in the inverted hierarchy withm 3 = 0 .1 eV are almost the same. The reason is simply that these two cases belong to the nearlydegenerate mass hierarchy (i.e., m 1 m 2 m 3 ).

The behaviors of B (H ++ ++ ) changing with the lightest neutrino mass are essentially similar to

those of

|m

| shown in Figure 4, and hence we do not go into detail in this connection.

4 CP violation in neutrino-antineutrino oscillations

In this section we carry out a detailed analysis of all the possible CP-violating asymmetries A between and oscillations. The generic expression of A has been given in Eq. (12). Becauseof the fact | m 231 | | m 232 | 32 m 221 , there may exist two oscillating regions dominated respectivelyby m 221 and m 231 . Let us make some analytical approximations for each of these two regions.

The oscillating region dominated by m 231 (or m 232 ), in which |31 |O(1) and 21 O(1). In

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A e e

1.0

0.5

0.0

0.5

1.0

A

1.0

0.5

0.0

0.5

1.0

A

L/E (m/keV)

1.0

0.5

0.0

0.5

1.0

0.1 1 10

A e

A e

A

0.1 1 10

Figure 8: The CP-violating asymmetries A versus L/E in the normal neutrino mass hierarchy withm 1 = 0, = 0 and = 45 .

the neglect of the m 221 -driven contributions, Eq. (12) approximates to

A31

2m 3 m 1 V 13 + m 2 V

23 sin231

m 2

4m 3 m 1 C13 + m 2 C

23 sin

2 31, (35)

where 32 31 has been taken into account. The oscillating region dominated by m 221 , in which 21 O(1) and |31 | O(1). Hence the

sin2

31 and sin2

32 terms in Eq. (3) oscillate too fast and each of them averages out to 1 / 2, whilethe sin 231 and sin 232 terms average out to zero. In this case Eq. (12) approximates to

A21

2m 1 m 2V 12 sin 221

m 2

2m 3 m 1C13 + m 2 C

23 4m 1 m 2 C

12 sin

2 21. (36)

Our numerical calculations will be based on the exact formula given in Eq. (12), but the approximationsmade in Eqs. (35) and (36) are helpful to understand the quantitative behaviors of A

ij . To reveal the

salient features of all the Aij , we are going to examine their dependence on the ratio L/E , the three

CP-violating phases and the absolute neutrino mass in Figures 816.

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A e e

1.0

0.5

0.0

0.5

1.0

A

1.0

0.5

0.0

0.5

1.0

A

L/E (m/keV)

1.0

0.5

0.0

0.5

1.0

0.1 1 10

A e

A e

A

0.1 1 10


4.1 The dependence of Aij on L/E and (,,)Let us consider three special cases for the neutrino mass spectrum, in which the expressions of A

ij can

be more or less simplied, to illustrate their dependence on the ratio L/E and the phases , and .(A) The normal neutrino mass hierarchy with m1 = 0. In this case we obtain m2 = m 221 8.66 10 3 eV and m 3 = m 231 4.97 10 2 eV. Eq. (12) is now simplied to

A =

2V 23 sin 231

m221

m 231 C22 + m

231

m 221 C33 + 2C

23 cos 231

. (37)

Note that the Majorana phase does not contribute to A in the m1 = 0 limit [6]. This point canalso be seen in Eq. (37): both C23 and V 23 do not contain , nor do C22 and C33 . For simplicity, wechoose (, ) = (0 , 45 ), (90 , 0 ) and (90 , 45 ) to calculate all the six independent Aij , and showtheir numerical results in Figures 810. Some comments are in order.

Figure 8 illustrates that signicant CP-violating effects can show up in neutrino-antineutrinooscillations even though the Dirac phase vanishes. They arise from the Majorana phase ,

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A e e

1.0

0.5

0.0

0.5

1.0

A

1.0

0.5

0.0

0.5

1.0

A

L/E (m/keV)

1.0

0.5

0.0

0.5

1.0

0.1 1 10

A e

A e

A

0.1 1 10


which has nothing to do with CP and T violation in normal neutrino-neutrino or antineutrino-antineutrino oscillations. We have taken = 45 to maximize each CP-violating term in A ,where and enter in the form of 2 and 2, as one can see from Eqs. (19)(24) 4 .

Figure 9 illustrates the nontrivial role of in generating CP or T violation in neutrino-antineutrinooscillations. Hence it is intrinsically a Majorana phase. In particular, = 90 (the most favoredvalue to enhance the magnitude of J ) can lead to large CP-violating asymmetries between e and e oscillations and between e and e oscillations. But the other four CP-violating asymmetries are quite insensitive to in this case.

A comparison between Figures 9 and 10 tells us again how important the Majorana phase isin producing CP and T violation. The interplay of and can be either positive or negative,depending on their explicit values. In order to determine all the three CP-violating phases, onehas to try to measure the CP-violating effects in as many channels as possible. Fortunately, notall the channels are strongly suppressed in most cases, unless , and themselves are too small

4 One may redene / 2 and / 2 in the PMNS matrix U , so as to eliminate the factor 2 from Eqs. (19)(24).

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A e e

1.0

0.5

0.0

0.5

1.0

A

1.0

0.5

0.0

0.5

1.0

A

L/E (m/keV)

1.0

0.5

0.0

0.5

1.0

0.1 1 10

A e

A e

A

0.1 1 10

Figure 11: The CP-violating asymmetries A versus L/E in the inverted neutrino mass hierarchy withm 3 = 0: (a) = 0 and = 45 (red dashed lines); (b) = 90 and = 0 (blue solid lines).

or take too special values.

When L/E O(1) m/keV, all the CP-violating asymmetries are averaged out to zero in this specialnormal mass hierarchy. Hence a measurement of A should better be done at L/E O(1) m/keV. Aproper arrangement of L/E may also maximize the signals of CP and T violation.

(B) The inverted neutrino mass hierarchy with m3 = 0. In this case we have m2 = m 232 4.93 10 2 eV and m 1 =

m 221 m 232 4.85 10 2 eV. Eq. (12) is then simplied to

A =2V 12 sin 221 m 221 + m 232 m 232 C11 + m 232 m 221 + m 232 C22 + 2C12 cos 221

. (38)

Because of m 221 | m 232 |, the coefficients of C11 and C22 in Eq. (38) are almost equal to one. Sothe dependence of A on these two mass-squared differences is rather weak. Note that all the Ado not depend on the absolute values of and in the m3 = 0 limit, but they depend on and . To illustrate, we typically choose ( , ) = (90 , 0 ) and (0 , 45 ) to calculate the CP-violatingasymmetries A . The numerical results are shown in Figure 11. Some discussions are in order.

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A e e

1.0

0.5

0.0

0.5

1.0

A

1.0

0.5

0.0

0.5

1.0

A

L/E (m/keV)

1.0

0.5

0.0

0.5

1.0

0.1 1 10

A e

A e

A

0.1 1 10

Figure 12: The CP-violating asymmetries A (blue solid lines) versus L/E in the nearly degenerateneutrino mass hierarchy with m1 m 2 m 3 , = = 0 and = 90 , where the red dashed lines standfor A21 in Eq. (40) with the oscillations driven by m 231 and m 232 being averaged out.

We see again that switching off the Dirac phase cannot forbid CP and T violation in neutrino-antineutrino oscillations. Instead, nontrivial values of may give rise to signicant CP-violatingeffects in all the channels under discussion.

Switching off can only lead to Aee = 0, simply because V 12ee = 0 holds in this case. Thanks tothe Dirac phase , large CP-violating asymmetries between and oscillationsare possible to show up.

In either case it is possible to achieve the so-called maximal CP violation (i.e., |A | = 1). Forexample, |Ae | 1 and |Ae | 1 can be obtained for proper values of L/E . Even |A | mayreach its maximal value at a suitable point of L/E [6].

In general, both and are the sources of CP and T violation. Since is always associated withs13 , its contribution to A is somewhat suppressed as compared with the contribution from . Thispoint can be clearly seen in Eqs. (19)(24). Nevertheless, the interplay of and sometimes playsthe dominant role in determining the size of A .

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A e e

1.0

0.5

0.0

0.5

1.0

A

1.0

0.5

0.0

0.5

1.0

A

L/E (m/keV)

1.0

0.5

0.0

0.5

1.0

0.1 1 10

A e

A e

A

0.1 1 10

Figure 13: The CP-violating asymmetries A (blue solid lines) versus L/E in the nearly degenerateneutrino mass hierarchy with m1 m 2 m 3 , = 0 , = 45 and = 90 , where the red dashed linesstand for A21 in Eq. (40) with the oscillations driven by m 231 and m 232 being averaged out.

(C) The nearly degenerate mass hierarchy with m1 m 2 m 3 . In this case m i m j can be factoredout and thus canceled on the right-hand side of Eq. (12), leading to the approximate expressions

A 2

i


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A e e

1.0

0.5

0.0

0.5

1.0

A

1.0

0.5

0.0

0.5

1.0

A

L/E (m/keV)

1.0

0.5

0.0

0.5

1.0

0.1 1 10

A e

A e

A

0.1 1 10

Figure 14: The CP-violating asymmetries A (blue solid lines) versus L/E in the nearly degenerateneutrino mass hierarchy with m1 m2 m3 , = = 45 and = 90 , where the red dashed linesstand for A21 in Eq. (40) with the oscillations driven by m 231 and m 232 being averaged out.

corresponding to the oscillating regions dominated by m 231 (or m 232 ) and m 221 , respectively. Notethat A31 are sensitive to all the three CP-violating phases, but only the phase difference and theDirac phase affect A21 . For the purpose of illustration, we typically take ( ,, ) = (0 , 0 , 90 ),(0 , 45 , 90 ) and (45 , 45 , 90 ) to calculate A . The numerical results are given in Figures 1214 5 .Some comments and discussions are in order.

Figure 12 illustrates the CP-violating effects in neutrino-antineutrino oscillations induced purelyby the Dirac phase . We see that Aee = 0 holds in this case, simply because the input = 90 istoo special to generate nonvanishing V 13ee and V 23ee , as shown in Eq. (19). When L/E is sufficientlylarge, the m 231 - and m 232 -dominated terms oscillate too fast and the observable behaviors of

A are essentially described by A21 . Once again we conclude that the CP-violating asymmetriesAe and Ae are most sensitive to . The same observation is true for the CP-violating asymmetrybetween normal e (or e ) and e (or e ) oscillations.

5 We have simply assumed the normal mass hierarchy and input m1 = 0 .1 eV in our numerical calculations. We ndthat the relevant results are almost the same if the inverted mass hierarchy with m 3 = 0 .1 eV is taken into account.

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A e e

1 . 0

0 . 5

0 . 0

0 . 5

1 . 0

A

1 . 0

0 . 5

0 . 0

0 . 5

1 . 0

A

m 1 (eV)

1 . 0

0 . 5

0 . 0

0 . 5

1 . 0

10 4 10 3 10 2 10 1

A e

A e

A

10 4 10 3 10 2 10 1

Figure 15: The CP-violating asymmetries A versus the lightest neutrino mass m1 in the normalhierarchy with = 45 , = 0 and = 90 , where the blue solid lines and red dashed lines correspondto L/E 0.25 m/keV and 8 m/keV, respectively.

Figure 13 illustrates the interplay of and in generating CP and T violation in neutrino-antineutrino oscillations. The suppressed CP-violating asymmetries in Figure 12 (because of = = 0 ) are now enhanced to a large extent. When both and are switched on, as shownin Figure 14, the situation becomes somewhat more complicated. In either case it is possible toachieve signicant or even maximal CP-violating asymmetries. In the oscillating region dominatedby m 231 and m 232 , the rst maximum or minimum of

A should be a good place to be detected.

The rst maximum or minimum of A in the m 231 -dominated oscillating region roughly occursaround L/E 0.25 m/keV, which corresponds to 31 / 4. In comparison, the rst maximum orminimum of A in the m 221 -dominated oscillating region may happen around L/E 8 m/keV,corresponding to 21 / 4. Of course, these results are more or less subject to the chosen inputs.

The above examples have illustrated the dependence of A on the ratio L/E and the three CP-violating phases in three special cases of the neutrino mass spectrum. In the subsequent subsection weshall examine the sensitivity of A to the absolute neutrino mass scale in a more careful way.

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A e e

1 . 0

0 . 5

0 . 0

0 . 5

1 . 0

A

1 . 0

0 . 5

0 . 0

0 . 5

1 . 0

A

m 1 (eV)

1 . 0

0 . 5

0 . 0

0 . 5

1 . 0

10 4 10 3 10 2 10 1

A e

A e

A

10 4 10 3 10 2 10 1

Figure 16: The CP-violating asymmetries A versus the lightest neutrino mass m3 in the invertedhierarchy with = 45 , = 0 and = 90 , where the blue solid lines and red dashed lines correspondto L/E 0.25 m/keV and 8 m/keV, respectively.

4.2 The sensitivity of A to m1 or m3To simplify our numerical calculations, we typically choose L/E 0.25 m/keV (i.e., 31 / 4) and8 m/keV (i.e., 21 / 4) which correspond to the m 231 - and m 221 -dominated oscillating regions,respectively. We also x = 45 , = 0 and = 90 to see the changes of A with the lightestneutrino mass m1 (normal hierarchy) or m3 (inverted hierarchy). The numerical results are shown in

Figures 15 and 16. Some comments are in order.

In Figure 15 the values of m1 change from O(10 4 ) eV to O(10 1 ) eV, implying the changesof the neutrino mass spectrum from m1 m2 m3 to m1 m2 m3 . The turning point isroughly m1 m 231 , around which the sensitivity of A to m1 becomes stronger. In the chosenparameter space we nd that Aee , Ae and Ae are most sensitive to m 1 at L/E 8 m/keV: theresults of these three CP-violating asymmetries for m1 0.1 eV are signicantly different fromthe ones for m1 0.

In Figure 16 the values of m3 change from O(10 4 ) eV to O(10 1 ) eV, implying the changes of the

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neutrino mass spectrum from m3 m1 m2 to m3 m1 m2 . The turning point is roughlym 3 | m 232 |, around which the sensitivity of A to m3 becomes more appreciable. But acomparison between Figures 15 and 16 tells us that the CP-violating asymmetries are in generalless sensitive to the absolute neutrino mass scale in the case of the inverted hierarchy.

As for the results of A , it does not make much difference whether the nearly degenerate neutrinomass spectrum is m1 m2 m3 or m3 m1 m2 . This point can be clearly seen in Figures15 and 16 at m 1 m 3 0.1 eV, where the numerical results in these two nearly degenerate massspectra approximately match each other.

So it is in principle possible to probe the absolute neutrino mass scale through the study of neutrino-antineutrino oscillation. In comparison, the normal neutrino-neutrino and antineutrino-antineutrinooscillations are only sensitive to the neutrino mass-squared differences.

5 Summary

One of the fundamental questions about massive neutrinos is what their nature is or whether they are theDirac or Majorana particles. The absolute neutrino mass scale is so low that it is extremely difficult todistinguish between the Dirac and Majorana neutrinos in all the currently available experiments. Todaystechniques have allowed us to push the sensitivity of the 0 decay to the level of | m ee |O(0.1) eV,making it the most feasible way to probe the Majorana nature of massive neutrinos. The present workis just motivated by a meaningful question that we have asked ourselves: what can we proceed to doto determine all the CP-violating phases in the PMNS matrix U if the massive neutrinos are somedayidentied to be the Majorana particles through a convincing measurement of the 0 decay?

In principle, one may determine the Majorana phases of U in neutrino-antineutrino oscillations 6 .In practice, such an experiment might only be feasible in the very distant future. But we nd that a

systematic study of CP violation in neutrino-antineutrino oscillations is still useful, so as to enrich thephenomenology of Majorana neutrinos. In this work we have explored the salient features of three-avorneutrino-antineutrino oscillations and their CP- and T-violating asymmetries. Six independent 0 -like mass terms m and nine independent Jalskog-like parameters V ij have been analyzed in detail,because they are quite universal and can contribute to the CP-conserving and CP-violating parts of anumber of LNV processes. We have made a comparison between V

ij and the Jarlskog invariant J by

switching off the Majorana phases and , and have demonstrated the Majorana nature of the Diracphase . As a by-product, the effects of three CP-violating phases on the LNV decays of doubly-and singly-charged Higgs bosons have also been reexamined. We have carried out a comprehensiveanalysis of the sensitivities of six possible CP-violating asymmetries A to the three phase parameters,the neutrino mass spectrum and the ratio of the neutrino beam energy E to the baseline length L.Our analytical and numerical results provide a complete description of the distinct roles of MajoranaCP-violating phases in neutrino-antineutrino oscillations and other LNV processes.

Although the particular parametrization of U advocated by the Particle Data Group [ 4] has been usedin this work, one may always choose a different representation of U which might be more convenient

6 Because the Dirac neutrinos do not violate the lepton number, they cannot undergo neutrino-antineutrino oscillations.However, it is likely for the Dirac neutrinos to oscillate between their left-handed and right-handed states in a magneticeld and in the presence of matter effects [25]. Such spin-avor precession processes are beyond the scope of the presentpaper and will be further studied elsewhere.

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in some aspects of the Majorana neutrino phenomenology. For instance, the so-called symmetricalparametrization of the PMNS matrix [26]

K =c12 c13 s12 c13 e i12 s13 e i13

s12 c23 ei12 c12 s13 s23 ei(23 13 ) c12 c23 s12 s13 s23 e

i(12 + 23 13 ) c13 s23 e i23

s12 s23 ei(12 + 23 ) c12 s13 c23 ei13 c12 s23 ei23 s12 s13 c23 e

i(12 13 ) c13 c23

(41)

has also been used by some authors to describe neutrino oscillations and LNV processes [27]. It is easyto establish the relationship between U in Eq. (1) and K in Eq. (41):

U =ei 0 00 ei 00 0 1

K , (42)

where the three phase parameters of U are related to the three phase parameters of K as follows:

= 13 12 23 , = 12 + 23 , = 23 . (43)

Therefore, it is straightforward to reexpress Cij and V ij in terms of the angle and phase parameters of K simply with the help of Eq. (43). Given three light or heavy sterile neutrinos, it is also straightforwardto extend Eq. (41) to a full parametrization of the 6 6 neutrino mixing matrix [ 28].

The Schechter-Valle (Black Box) theorem [29] has told us that an observation of the 0 decaypoints to the Majorana nature of massive neutrinos, but such a LNV process may be dominated eitherby a tree-level Majorana neutrino mass term or by other possible new physics which is essentiallyunrelated to the neutrino masses. The radiative mass term induced by the Black Box (loop) diagram

itself is extremely small in most cases, although this is not always true [30]. Hence one has to be carefulwhen relating the rate of the 0 decay fully to the neutrino masses. In this work we have assumedthe existence of a tree-level Majorana mass term dominating the Black Box diagram, leading to m eewhich has a direct relation to the rate of the 0 decay. The same observation is expected to be truefor all the LNV processes which depend on the effective Majorana mass terms m . Of course, thesituation will change if other types of LNV physics exist [ 30].

While it is still a dream to fully determine the avor dynamics of Majorana neutrinos, includingtheir CP-violating phases, one should not be too pessimistic. The reason is simply that the history of neutrino physics has been full of surprises in making the impossible possible, but one has to be patient.

Acknowledgement

We would like to thank Y.F. Li for helpful discussions. This work was supported in part by theNational Natural Science Foundation of China under Grant No. 11135009.

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A Explicit expressions of Cij Given the standard parametrization of the PMNS matrix U in Eq. (1), one may explicitly write out allthe CP-conserving quantities Cij dened in Eq. (5). Such formulas are expected to be useful to under-stand the behaviors of neutrino-antineutrino oscillations and make reasonable analytical approximationsfor their oscillation probabilities and CP-violating asymmetries.

First of all, we have C ii = |U i |4 (for = e, , and i = 1 , 2, 3). The explicit expressions of thesenine quantities are

C11ee = c

412 c

413 ,

C22ee = s

412 c

413 ,

C33ee = s

413 ;

C11 = s

212 c

223 + 2 c12 s12 s13 c23 s23 cos + c

212 s

213 s

223

2,

C22 = c

212 c

223 2c12 s12 s13 c23 s23 cos + s

212 s

213 s

223

2,

C33 = c

413 s

423 ;

C11 = s212 s223 2c12 s12 s13 c23 s23 cos + c212 s213 c223 2 ,C

22 = c

212 s

223 + 2 c12 s12 s13 c23 s23 cos + s

212 s

213 c

223

2,

C33 = c

413 c

423 . (44)

Because Cii = Cii Cii holds, it is straightforward to write out the expressions of Cii (for = )with the help of Eq. (44). The following sum rule is also valid:

Cii =

|U i |2 = 1 . (45)

We see that all the Cii are independent of the Majorana phases and .Next, we calculate C

ij and C

ij in terms of the avor mixing parameters of the PMNS matrix U

given in Eq. (1). The results are

C12ee = c

212 s

212 c

413 cos 2 ( ) ,

C13ee = c

212 c

213 s

213 cos 2 ( + ) ,

C23ee = s

212 c

213 s

213 cos 2 ( + ) ;

C12 = c

212 s

212 c

423 4s

213 c

223 s

223 + s

413 s

423 cos2( )

+2 c12 s12 s13 c23 s23 c223 s

213 s

223 c

212 cos (2 2 + ) s

212 cos (2 2 )

+ s213

c223

s223

c412

cos 2 (

+ ) + s4

12 cos 2 (

) ,

C13 = c

213 s

223 s

212 c

223 cos 2 + 2 c12 s12 s13 c23 s23 cos ( + 2 ) + c

212 s

213 s

223 cos 2 ( + ) ,

C23 = c

213 s

223 c

212 c

223 cos 2 2c12 s12 s13 c23 s23 cos ( + 2 ) + s

212 s

213 s

223 cos 2 ( + ) ;

C12 = c

212 s

212 s

423 4s

213 c

223 s

223 + s

413 c

423 cos2( )

2c12 s12 s13 c23 s23 s223 s

213 c

223 c

212 cos (2 2 + ) s

212 cos (2 2 )

+ s213 c223 s

223 c

412 cos 2 ( + ) + s

412 cos 2 ( ) ,

C13 = c

213 c

223 s

212 s

223 cos 2 2c12 s12 s13 c23 s23 cos ( + 2 ) + c

212 s

213 c

223 cos 2 ( + ) ,

C23 = c

213 c

223 c

212 s

223 cos 2 + 2 c12 s12 s13 c23 s23 cos ( + 2 ) + s

212 s

213 c

223 cos 2 ( + ) ; (46)

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The lower and upper bounds of | m | turn out to be

m 221 s212 m 231 s213 | m ee | m 221 s212 + m 231 s213 , m 231 s223 m 221 c212 c223 m m 231 s223 + m 221 c212 c223 ,

m 231 c223

m 221 c

212 s

223

|m

|

m 231 c

223 +

m 221 c

212 s

223 ,

m 231 s13 s23 m 221 c12 s12 c23 m e m 231 s13 s23 + m 221 c12 s12 c23 , m 231 s13 c23 m 221 c12 s12 s23 | m e | m 231 s13 c23 + m 221 c12 s12 s23 , m 231 c23 s23 m 221 c212 c23 s23 m m 231 c23 s23 + m 221 c212 c23 s23 . (50)We see that the dominant terms of | m |, | m | and | m | are associated with m 231 , and theyreceive some small corrections from the terms proportional to m 221 . In comparison, the magnitudeof | m ee | is much smaller because its m 231 term is suppressed by s213 . The dominant terms of | m e |and | m e | are associated with

m 231 s13 , and thus they are somewhat less suppressed.

Case (B): the lightest neutrino mass m3 satises m23 m

221 . In this case, m1 m 2 m

232

holds. We are allowed to neglect those terms proportional to m3 in Eq. (27). Given the smallness of 13 , we approximately obtain

m ee + m 232 c212 e2i + s212 e2i ,m + m 232 c223 s212 e2i + c212 e2i ,m + m 232 s223 c212 e2i s212 e2i ,m e +

m 232 c12 s12 c23 e

2i e2i ,

m e + m232 c12 s12 s23 e2i e2i ,

m m 232 c23 s23 s212 e2i + c212 e2i . (51)Then the lower and upper bounds of | m | are approximately given by

m 232 cos212 | m ee | m 232 , m 232 c223 cos 212 m m 232 c223 ,

m 232 s

223 cos 212 | m |

m 232 s

223 ,

0 m e m232 sin212 c23 ,

0 | m e | m 232 sin212 s23 , m 232 c23 s23 cos 212 m m 232 c23 s23 . (52)We see that the lower bounds of | m e | and | m e | are zero, while the allowed ranges of the other foureffective mass terms are quite narrow and at the level of m 232 . Given , even the texture zerosm e m e 0 can be achieved. A systematic analysis of such two-zero textures of the Majorananeutrino mass matrix M has been done in the literature [ 31].

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Case (C): the lightest neutrino mass m1 satises m21 m 231 . This case corresponds to a nearlydegenerate mass hierarchy: m1 m2 and m3 m1 + m 231 / (2m 1 ). Given the smallness of 13 , theexpressions of m in Eq. (27) approximate to

m ee m1 c212 e

2i + s212 e2i ,

m

m1 s223 + s

212 c

223 e

2i + c212 c223 e

2i + m 231

2m 1s223 ,

m m1 c223 + c

212 s

223 e

2i + s212 s223 e

2i + m 231

2m 1c223 ,

m e m1 c12 s12 c23 e2i e

2i ,

m e m1 c12 s12 s23 e2i e

2i ,

m m1 c23 s23 1 c212 e

2i s212 e

2i + m 231

2m 1c23 s23 . (53)

The lower and upper bounds of | m | turn out to be

m 1 cos212 | m ee | m 1 ,0 m m 1 ,m 1 cos 223 +

m 2312m 1

c223 | m | m 1 ,0 m e m 1 sin 212 c23 ,

0 | m e | m 1 sin 212 s23 , m 2314m 1

sin223 m m 1 sin 223 . (54)

Note that the lower bounds of | m | and | m | obtained above hold only in the 23 < 45 case,consistent with the numerical calculations done in section 3.2. If 23 > 45

is supported by the future

experimental data, then the lower bounds of | m | and | m | in Eq. (54) should be exchanged.Case (D): the lightest neutrino mass m3 satises m23 | m 232 |. This case also corresponds to a

nearly degenerate mass hierarchy: m1 m2 and m3 m1 + m 232 / (2m 1 ), where m 232 < 0. Theapproximate expressions of m can similarly be obtained as in Eq. (53) by replacing m 231 with m 232 . Then we arrive at

m 1 cos212 | m ee | m 1 ,0 m m 1 ,

m 1 cos 223 + m 232

2m 1c223

|m

| m 1 ,

0 m e m 1 sin 212 c23 ,

0 | m e | m 1 sin 212 s23 ,0 m m 1 sin 223 . (55)

It is worth pointing out that the lower bound of | m | given in Eq. (51) or Eq. (52) should bezero if 23 is nally found to lie in the second quadrant and makes m1 cos223 + m 231 c223 / (2m 1 ) orm 1 cos 223 + m 232 c223 / (2m 1 ) negative.

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[16] N. Cabibbo, Phys. Lett. B 72 , 333 (1978).

[17] Z.Z. Xing, Phys. Lett. B 633 , 550 (2006); Z.Z. Xing and H. Zhang, Commun. Theor. Phys. 48 , 525(2007).

[18] E. Komatsu et al. (WMAP Collaboration), Astrophys. J. Suppl. 192 , 18 (2011).

[19] P.A.R. Ade et al. (Planck Collaboration), arXiv:1303.5076.

[20] Z.Z. Xing, Phys. Rev. D 68 , 053002 (2003); Y.F. Li and S.S. Liu, Phys. Lett. B 706 , 406 (2012).

[21] See, e.g., M. Frigerio and A.Yu. Smirnov, Phys. Rev. D 67 , 013007 (2003); Z.Z. Xing, Phys. Rev.D 69 , 013006 (2004); A. Merle and W. Rodejohann, Phys. Rev. D 73 , 073012 (2006); S. Zhou andJ.Y. Liu, Phys. Rev. D 87 , 093010 (2013).

[22] See, e.g., Z.Z. Xing and S. Zhou, Neutrinos in Particle Physics, Astronomy and Cosmology (Zhe- jiang University Press and Springer-Verlag, 2011).

[23] See, e.g., J. Garayoa and T. Schwetz, JHEP 0803 , 009 (2008); M. Kadastik, M. Raidal, and L.Rebane, Phys. Rev. D 77 , 115023 (2008); A.G. Akeroyd, M. Aoki, and H. Sugiyama, Phys. Rev. D77 , 075010; Z.Z. Xing, Phys. Rev. D 78 , 011301 (2008); P. Fileviez Perez, T. Han, G.Y. Huang, T.Li, and K. Wang, Phys. Rev. D 78 , 015018 (2008); F. del Aguila and J.A. Aguilar-Saavedra, Nucl.Phys. B 813 , 22 (2009).

[24] P. Ren and Z.Z. Xing, Phys. Lett. B 666 , 48 (2008); Chin. Phys. C 34 , 433 (2010).

[25] For a recent review with extensive references, see: C. Giunti and A. Studenikin, Phys. Atom. Nucl.72 , 2089 (2009); and references therein.

[26] J. Schechter and J.W.F. Valle, Phys. Rev. D 22 , 2227 (1980).

[27] W. Rodejohann and J.W.F. Valle, Phys. Rev. D 84 , 073011 (2011).

[28] Z.Z. Xing, Phys. Lett. B 660 , 515 (2008); Phys. Rev. D 85 , 013008 (2012).

[29] J. Schechter and J.W.F. Valle, Phys. Rev. D 25 , 2951 (1982).

[30] M. Duerr, M. Lindner, and A. Merle, JHEP 1106 , 091 (2011).

[31] See, e.g., Z.Z. Xing, Phys. Lett. B 530 , 159 (2002); Phys. Lett. B 539 , 85 (2002); P.H. Frampton,S.L. Glashow, and D. Marfatia, Phys. Lett. B 536 , 79 (2002); W.L. Guo and Z.Z. Xing, Phys. Rev.D 67 , 053002 (2003); H. Fritzsch, Z.Z. Xing, and S. Zhou, JHEP 1109 , 083 (2011); T. Araki, J.Heeck, and J. Kubo, JHEP 1207 , 083 (2012).
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