CP violations in predictive neutrino mass

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Web End = CP violations in predictive neutrino mass structures

Chao-Qiang Geng1,2,3,a, Da Huang2,b, Lu-Hsing Tsai2,c

1 Chongqing University of Posts and Telecommunications, Chongqing 400065, China

2 Department of Physics, National Tsing Hua University, Hsinchu, Taiwan

3 Physics Division, National Center for Theoretical Sciences, Hsinchu, Taiwan

Received: 13 September 2015 / Accepted: 5 November 2015 / Published online: 26 November 2015 The Author(s) 2015. This article is published with open access at Springerlink.com

Abstract We study the CP-violation effects from two types of neutrino mass matrices with (i) (M)ee = 0,

and (ii) (M)ee = (M)e = 0, which can be realized

by the high-dimensional lepton number violating operators

cR LL (D ) 2 and cRlR(D )2 2, respectively. In (i),

the neutrino mass spectrum is in the normal ordering with the lightest neutrino mass within the range 0.002 eV [lessorsimilar] m0 [lessorsimilar]

0.007 eV. Furthermore, for a given value of m0, there are two solutions for the two Majorana phases 21 and 31, whereas the Dirac phase is arbitrary. For (ii), the parameters of m0, , 21, and 31 can be completely determined. We calculate the CP-violating asymmetries in neutrinoantineutrino oscillations for both mass textures of (i) and (ii), which are closely related to the CP-violating Majorana phases.

1 Introduction

Although it has been established that neutrinos are massive and mix each other in the recent several decades [17], their nature is still mysterious. It is known that neutrino mass terms could be of the Dirac type, in analogy to the charged fermions,i.e. quarks and charged leptons, or the Majorana type, possibly generated by the Weinberg operator LcL [8]. In

the literature, there have been many models to realize the Weinberg operator at the tree [921] and loop [2228] levels. Note that the current neutrino oscillation experiments cannot determine the three CP-violating phases, especially for the two Majorana phases, which is an important problem in neutrino physics.

The Weinberg operator violates the lepton number symmetry by two units, but sometimes it is not the one that gives the dominant contribution to the Majorana neutrino masses or the lepton number violating (LNV) processes.

a e-mail: mailto:[email protected]

Web End [email protected]

b e-mail: mailto:[email protected]

Web End [email protected]

c e-mail: mailto:[email protected]

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Instead, other higher-dimensional LNV operators, for example, the dimension-7, O7 =

cRLL (D ) 2 [2932] and dimension-9, O9 =

cR R(D )2 2 [3338] operators, can lead to new Majorana neutrino mass structures different from those by the Weinberg operator if they are prominent. Specifically, due to the non-trivial dependence of the charged lepton masses, O7 generically generates a neutrino mass matrix

with (M)ee = 0 [32] in the avor basis, while O9 nat

urally gives rise to the texture with (M)ee = (M)e =

0 [28,29,33,35,37,38]. By tting the present neutrino oscillation data, both textures predict that the neutrino mass matrix should be of the normal ordering, and already give stringent constraints to the unknown parameters in the neutrino mass matrix. In particular, in Refs. [32,38], we show that they could naturally lead to non-trivial values for the three CP-violating phases. We regard that the higher-dimensional operators would provide us with a new way to generate the new neutrino structures, besides the ordinary approach by imposing avor symmetries [3949].

In the present paper, we investigate a relevant question: to what extent can the conditions (M)ee = 0 and (M)ee =

(M)e = 0 restrict the neutrino mass matrix structure, espe

cially the leptonic CP-violating phases, based on measured quantities from oscillation experiments? In our treatment, we also take into account the experimental uncertainties in the data in order to see their effects on the results. Note that there have already been many studies of these two specic neutrino mass matrices in the literature; see e.g., Refs. [50 67], in which more texture-zero neutrino mass matrices were examined. Here, our focus is their implications on the lep-tonic CP-violating phases.

With the predicted neutrino mass parameters, the next question is how to test the above two texture-zero structures by measuring all the relevant parameters in the neutrino mass matrices, especially the non-trivial Majorana phases. Previous studies showed that neutrinoantineutrino oscillations gave us prospective approaches to probe the Majo-

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rana phases [6877], which is impossible for the conventional (anti)neutrino(anti)neutrino oscillation experiments.We nd that once the possible regions of these phases are depicted for the present two textures, the associated CP-violating asymmetries of the neutrinoantineutrino oscillations can be predicted. As will be shown later, by appropriately choosing the (anti)neutrino beam energy and baseline length, some of the asymmetries can be of O(1).

This paper is organized as follows. In Sect. 2 we study the implications of the texture-zero conditions (M)ee = 0

and (M)ee = (M)e = 0 to the unknown neutrino mass

parameters, including the lightest neutrino mass and the three CP-violating phases, based on the existing data. With the preferred values of these parameters, we predict the CP-violating asymmetries in the neutrinoantineutrino oscillations for both textures in Sect. 3. In Sect. 4, we give a short summary.

2 Texture-zero neutrino mass matrix

As the Majorana neutrino mass matrix M is symmetric, there are six independent complex elements (M)ee, (M)e, (M)e , (M), (M) , and (M) . A well-dened M can be connected with the observed quantities from neutrino oscillations. Up to the eld redenition, all of the above matrix elements depend on the nine neutrino parameters, including three masses, three mixing angles, one Dirac CP phase and two Majorana CP phases. In the avor basis, where the charged lepton mass matrix is diagonal, the neutrino mass matrix dened in the Lagrangian L = 12(L )cM L + H.c. can be decomposed as fol

lows:

M V diag(m1, m2, m3)V , (1) where m1,2,3 are three neutrino masses. V is the charged current leptonic mixing matrix [78,79], conventionally expressed in the standard parametrization as [80,81]

V =

with (m1, m2, m3) = m0, m20 + m221, m20 + m231 , and the inverted one for m232 < 0 with (m1, m2, m3) =

m20 m231, m20 m231 + m221, m0 .

M can have some special approximate texture-zero forms when it is generated by some high-dimensional LNV operators. For example, if O7 =

cR LL (D ) 2 gives the leading contribution to neutrino masses, then (M) should be approximately proportional to the sum of charged lepton masses, m + m , with and = e, , . Consequently,

(M)ee should be much smaller than other elements. Similarly, if O9 =

cR R(D )2 2 dominates over other LNV operators, (M) will be proportional to m m . It turns out that not only (M)ee but also (M)e are expected be greatly suppressed due to the hierarchy in the charged lepton masses. In other words, the neutrino mass matrices obtained from these LNV effective operators are characterized by the special zero textures (M)ee = 0 and (M)ee = (M)e = 0,1

the implications of which will be discussed in detail in the following two subsections.

2.1 (M)ee = 0

By expanding the right-hand side of Eq. (1) with the standard parametrization of V in Eq. (2), the condition (M)ee = 0

can be transformed into the following relation:

(M)ee = c212c213m1 + s212c213m2ei21 + s213m3ei = 0,(3)

where the phase 31 2 is dened, which will be

used to replace 31 as an independent Majorana phase hereafter. Note that this equation excludes the inverted ordering at more than 2 signicance by current oscillation experiment results [81], so that we only need to consider the normal ordering neutrino mass matrix from now on. Figure 1 shows the allowed parameter space region satisfying Eq. (3), in which the solid curves represent the parameters when the experimental observables are at their central values in PDG [81], while the shadow areas correspond to the 1 standard deviations. It is interesting to note that m0 and 21 are already limited within small parameter regions, with 0.8 [lessorsimilar] 21 [lessorsimilar] 1.2 and 0.0015 eV [lessorsimilar] m0 [lessorsimilar] 0.008 eV,

1 Besides the relative smallness of the element (M)ee already argued in the main text for the two high-dimensional effective operators of O7

and O9, its absolute value is further constrained by the neutrinoless

double beta (0) decay processes [8288]. Note that these two effective operators give the dominant contributions to the 0 decay at tree level, while the Majorana mass terms arising from O7(9) begins at one-

(two-)loop level. Due to the absence of the loop suppression, these two operators are more sensitive to the 0 decay processes, which constrain the cutoff scales and Wilson coefcients of the effective operators greatly and lead to the conclusion that (M)ee < 1013 eV. For further details, please refer to Refs. [32] and [38].

c12c13 s12c13 s13ei

s12c23 c12s23s13ei c12c23 s12s23s13ei s23c13

s12s23 c12c23s13ei c12s23 s12c23s13ei c23c13

1 0 0 0 ei21/2 0

0 0 ei31/2

, (2)

where si j sin i j, ci j cos i j, is the Dirac phase,

and 21,31 represent two Majorana phases within the range

[0, 2]. The values of 12, 23, 13, m221, and | m232| have

already been obtained from the neutrino oscillation experiments [81], so that the remaining four unknown neutrino parameters are the three CP phases, , 21, and 32, and the lightest neutrino mass, m0. Note that only the absolute value m232 has been acquired, which leaves us two possible orderings: the normal ordering for m232 > 0

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Eur. Phys. J. C (2015) 75 :557 Page 3 of 8 557

Fig. 1 Contours for(M)ee = 0 in a m021 and b

m0 planes, respectively.

Solid lines with experimental central values and 1 standard deviation, where the colors of red and blue indicate that the values of sin 21 are positive and negative, respectively

0.000 0.002 0.004 0.006 0.008 0.010

(a) (b)

2.0

1.5

1.0

0.5

0.0

0.000 0.002 0.004 0.006 0.008 0.010

m0 eV m0 eV

respectively. In particular, the extremal values of the lightest neutrino mass (m0)min(max) are related to two CP-conserving solutions to Eq. (3) with 21 = and = 0(), s212c213m2 c212c213m1 s213m3 = 0 andc212c213m1 s212c213m2 s213m3 = 0. (4)

For each value of m0 within the regions [(m0)min, (m0)max],

there exist two solutions for 21 and , differentiated by the positive or negative sin 21, which are shown in Fig. 1 as red or blue curves/shadows. Another interesting observation is that the obtained 21 is limited around , which can be understood directly from Eq. (3). Since s213 is very small, the third term in Eq. (3) can be neglected, and the rst two terms must balance each other to achieve the constraint of the vanishing (M)ee, which only requires 21 in order

to reverse the sign of the second term. Moreover, 21 is precisely predicted to be 1.1 or 0.9 when m0 is located within 0.004 eV [lessorsimilar] m0 [lessorsimilar] 0.005 eV,2 no matter how the experimental errors vary. Finally, we remark that (M)ee = 0 does

not provide any constraint on , which is only contained in . If one focuses on the real M, then can be taken as 0 or , for the cases m0 = (m0)min and (m0)max. Therefore,

there are four independent real neutrino mass matrices for (M)ee = 0.

2.2 (M)ee = (M)e = 0

In this subsection, we will concentrate on the case with (M)ee = (M)e = 0. Note that such constraints corre

spond to two complex equations, which enable us to uniquely solve for the remaining four parameters (m0, , 21, ) in the neutrino mass matrix undetermined from the current oscillation experiments. Now we sketch the procedure of deriving these quantities in terms of the measured observables [28]. The rst step is to write down the two conditions in the parametrization independent form:

(M)ee = m1V 2e1 + m2V 2e2 + m3V 2e3 = 0,(M)e = m1V e1V 1 + m2V e2V 2 + m3V e3V 3 = 0, (5)

2 Similar results are also given in Ref. [76].

with which we can obtain the following useful formulas:

m221

m231 =

1 |Y |2

1 |X|2

, (6)

m2m3 = |

Y | |X|

, (7)

Im(Ve3V3V e2V 2) = Im(Ve1V1V e3V 3)= Im(Ve2V2V e1V 1) = 0, (8)

with

X =

Ve2V2V e1V 1

|Ve1|2|Ve1V3 Ve3V1|2

. (9)

Since neither |X| nor |Y | depends on the two Majorana phases

and m0, we can determine the Dirac phase from Eq. (6). By substituting the obtained Dirac phase into Eq. (7), we can solve for m0. Finally, two Majorana phases can be xed with Eq. (8). In the standard parametrization, the solution is expressed by [28]

cos =

s113

2(1 + t212) + r(1 t212)

t12

t23 r(t213 1)

Y 1 =

Ve1V1V e3V 3

|Ve3|2|Ve3V2 Ve2V3|2

(1 t412)+r2(1 + t412) ,

(10a)

= arg(s13 + t12t23ei) , (10b)

21 = arg

s13 t12t23ei s13 + t112t23ei

t23

t12

, (10c)

m0 =

m221

(2 + r)

t213(t212t223 2s13t12t23c + s213) 1 t213(1 + t212t223 2s13t12t23c)

(10d)

with r m221/( m232 + m221/2). Note that for each value

of m0, we can obtain two solutions of the CP-violating phases (, 21, and ), which can be connected with each other by the replacements of 2 , 21 2 21, and

2 .

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557 Page 4 of 8 Eur. Phys. J. C (2015) 75 :557

0.01

0 0 0.5 1 1.5 2

0.01

0 0 0.5 1 1.5 2

0.01

0 0 0.5 1 1.5 2

0.008

0.006

m 0

0.004

0.002

1.5

0.5

0 0 0.5 1 1.5 2

1.5

0.5

0 0 0.5 1 1.5 2

Fig. 2 Correlations among the parameters: , , 21, and m0, where the red (blue) color represents that sin 21 > 0 (< 0)

Figure 2 shows the allowed parameter space according to Eq. (10) when we take the tting values of the ve parameters (12, 23, 13, m221, m32) within the 1 deviation as the input parameters, in which the red (blue) areas label the regions with sin 21 > 0(< 0). If we take the central experimental values of the measured quantities [81], two solutions can be obtained with m0 = 5.07 103 eV,

= 0.59 (1.41), 21 = 0.89 (1.11), and =

1.34 (0.66). The corresponding leptonic Jarlskog invariant, J = c12c213c23s12s13s23s, is equal to 0.033(0.033),

which characterizes CP-violation in the lepton sector. It is worth noting that the value of = 1.41 for one of the

solutions is close to the central value of from the global tting result [81]. Moreover, in the present texture-zero case, when the Dirac phase is taken to be the CP-conserving values, such as = 0 or , the two Majorana ones can only

be CP-conserving values too, i.e., 21 and should be 0 or . However, with the results shown in Fig. 2, it is interesting that the CP-conserving cases are excluded at the 1 level. Finally, if the Dirac phase is taken to be of the maximal CP-violating value with = /2 (/2), the Majo

rana phases are predicted to be 21 = 0.88 (1.12) and

= 1.42 (0.58) with the experimental central values for

the mixing angles.

3 Neutrinoantineutrino oscillations

With the non-trivial Majorana CP-violating phases, especially for the case with Mee = Me = 0, the immedi

ate important question is how to measure them. Traditional (anti)neutrino(anti)neutrino oscillation experiments can be used to measure the Dirac phase, but they are insensitive to the Majorana ones since the involved processes are lepton number conserving so that the Majorana phases are canceled out in the corresponding formulas. As a result, in order to measure the Majorana phases, one of the necessary conditions is that the involved processes are LNV. As pointed out in Refs. [6871], the neutrinoantineutrino oscillations provide us with a promising way to detect them. Unfortunately, the neutrinoantineutrino channels suffer from an additional helicity suppression factor (m/E)2 in the oscillation probabilities, compared with the corresponding usual (anti)neutrino(anti)neutrino oscillation channels. Therefore, it is challenging to carry out such experiments. The use of the low-energy Mssbauer electron antineutri

nos [8995] with 18.6 keV, which are emitted from the bound-state beta decay of 3H to 3He, can improve the situation greatly by enhancing the signal by a factor of O(104)

as compared with the conventional reactor antineutrinos [71]. However, even in this case, it is still practically impossible to

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Eur. Phys. J. C (2015) 75 :557 Page 5 of 8 557

observe these neutrinoantineutrino oscillations, as will be shown below.

The general formulas for the neutrinoantineutrino oscillation probabilities P(

) and P(

) in the

phenomenological interest, and then see how these measurements can help us to probe or constrain the whole picture of neutrino masses.

3.1 (M)ee = 0

According to Eq. (13), the asymmetry Aee can be expressed by

Aee = 2 m1m2c

2 12c4

13s2

12 sin2

three-avor framework are [71]

P( ) = |

K |

|M|

2 4

i< j

mi m j Re(Vi Vi V

j V

j )

sin2

m2ji L 4E

+ 2

i< j

mi m j Im(Vi Vi V

j V

j ) sin

m2ji L 2E

(11)

2 sin 21 + m1m3c

2 12c2

13s2

sin2 f 2

1 + m232 m221

sin

) = |

K |2

|M|

2 4

i< j

mim j Re(Vi Vi V j V j)

sin2

m2m3c213s212s213 sin2

f m232 2 m221

sin(21 )

m2ji L

i< j

mim j Im(Vi Vi V j V j) sin

m2ji L

|(M)ee|

+4m1m2c212c413s212 sin2

(12)

where K and K are the kinetic factors with |K | = | K | and

L is the neutrino traveling length. Now it is interesting to estimate the neutrinoantineutrino oscillation probabilities for different channels to see if they have the potential to be observed under the present experimental status, especially the Mssbauer neutrinos advertised in Ref. [71]. By assuming

the kinematic factor K O(1), electron antineutrino energy

E 18.6 keV, and oscillation baseline length L 300 m,

we can obtain the largest e

4 cos 21 + 4m1m3c

2 12c2

13s2

sin2

f 4

1 + m232 m221

cos

+4m2m3c213s212s213 sin2

f m232 4 m221

e oscillation probability to be P(e e) O(1013) for m0 = 0.0065 eV. The largest

probabilities for other oscillation channels, such as P(

cos(21 ) . (14)

When imposing the condition (M)ee = 0, both 21 and

can be expressed as the functions of m0. Therefore, by xing the factor f to some denite value, the CP-violating asymmetries of various channels can also have denite values for every m0. As an illustration, Fig. 3a gives the correlation between Aee and m0 when f = 0.55, where we only take

the central values of the measured quantities in our calculation. By comparing Figs. 3a and 1a, we see that the detection of Aee directly implies the existence of a non-zero sin 21, and their signs are positively correlated. Our direct calculation conrms this observation. We also plot the variation of Aee against the factor f in Fig. 3b by taking m0 = 0.004 eV,

which shows that if we can ne tune the beam energy E or

), would be of a similar order. In the view of these simple exercises, it seems impossible to observe these oscillations practically in the foreseeable experiments.

It is obvious that P( ) and its CP conjugate pro

cess P( ) can have different values when V is com

plex, which is the origin of CP-violation in the lepton sector. Therefore, we can dene the CP asymmetry parameter A

P( ) P(

)

P( ) + P(

)

2 i< j mim j Im(Vi Vi V j V j) sin ( f m2ji)/(2 m221)

= |M|2 4 i< j mimj Re(Vi Vi V j V j) sin2 ( f m2ji)/(4 m221)

(13)

where f = (L/E)( m221/). In the following, we shall

use the obtained CP-violating phases from the previous two texture-zero neutrino mass matrices to predict the oscillation probabilities and the associated CP-violating asymmetries in some neutrinoantineutrino oscillation channels of great

the baseline length L to make an appropriate value of f , a large CP-violating asymmetry in the e

e channel |Aee| 1

can be obtained. Furthermore, note that the dependence of Aee on the Dirac phase is only through the combination of = 312, so that even if vanishes, there can still be quite

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557 Page 6 of 8 Eur. Phys. J. C (2015) 75 :557

Table 1 Asymmetries for neutrinoantineutrino oscillations with f = 0.55 and m0 = 0.004 eV

Aee Ae Ae A A A

0 1.0 (1.0) 0.38 (0.38) 0.71 (0.71) 0.19 (0.19) 0.21 (0.21) 0.17 (0.17)

/2 1.0 (1.0) 0.61 (0.20) 0.20 (0.47) 0.26 (0.25) 0.18 (0.19) 0.15 (0.16)

Fig. 3 Aee as functions of a m0 and b f , where the red (blue) corresponds tosin 21 > 0 (< 0)

(a) (b)

1.0

0.5

A ee

0.0

A ee

0.0

0.5

1.0 0.0 0.2 0.4 0.6 0.8 1.0

1.0

0.000 0.002 0.004 0.006 0.008 0.010

m0 eV

sizable CP-violating effects in the e

e oscillation experiment due to the compensation from 31 in , which starkly shows the signicance of the Majorana phases in generating the CP-violating effects. Finally, we make the estimation of the asymmetries in other neutrinoantineutrino oscillation channels, by simply taking f = 0.55, m0 = 0.004 eV, and

= 0 or /2, together with other measured observables at

their central values, with the results shown in Table 1.

3.2 (M)ee = (M)e = 0

It was shown previously that all of the mass and mixing parameters can be xed when (M)ee = (M)e = 0. It fol

lows that all of the CP-violating asymmetries in the neutrino antineutrino oscillations can also be determined. We remark that for this case Ae always vanishes since each term in the summation of Eq. (13) is zero as the consequence of Eq. (8). Using the central values of neutrino mixing parameters from neutrino oscillations with f = 3.5, we can predict

Aee = 0.92, Ae = 0.09, A = 0.15, A = 0.1, and

A = 0.09. Therefore, the e

e oscillation is the most prospective channel to probe this neutrino mass texture.

4 Summary

We have studied the CP-violating asymmetries and related LNV processes such as the neutrinoantineutrino oscillations under two types of the neutrino mass textures, (M)ee = 0

and (M)ee = (M)e = 0, realized by the high-dimensional

lepton number violating operators. For (M)ee = 0, there

are two solutions of 21 and for each value of m0, with 0.002 eV [lessorsimilar] m0 [lessorsimilar] 0.007 eV and an arbitrary value of .

For (M)ee = (M)e = 0, two solutions for free parame

ters (m0, , 21, ) can be obtained, in which one of them, with = 1.41, is close to the global tting result. The

effect of the non-zero values of the two Majorana phases can be reected by the related CP-violating asymmetry parameters A in neutrinoantineutrino oscillations. In the texture (M)ee = 0, we nd that a non-zero Aee can be obtained even

if the Dirac phase is switched off, and its sign is positively correlated to that of sin 21. For (M)ee = (M)e = 0, a

large values of Aee is predicted, while Ae is always zero.

It is interesting to consider other probes of the Majorana character of the neutrino masses, such as rare LNV meson decays. It is well known that ordinary channels with Majorana neutrino mass insertions are too small to be observed in the near future. However, it is remarkable that the effective

Fig. 4 Leading-order Feynman diagrams for rare LNV B meson decays induced by a O9 and b

(a) (b)

W O7

O9 B

W +

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Eur. Phys. J. C (2015) 75 :557 Page 7 of 8 557

operators, such as O7 and O9, would give new leading-order

contributions. For concreteness, let us consider the process B+ ++. If Majorana neutrino masses are induced

by O9, the dominant channel to this process is given by the

Feynman diagram in Fig. 4a, as this tree-level diagram does not involve the tiny Majorana neutrino masses which arise at two-loop level via O9. However, with the model parameters

xed by the observed neutrino masses as in Ref. [38], a simple estimation shows that the typical branching ratio for this process is to be of O(1025). Other LNV rare meson decays,

like K + ++, would have even smaller branch

ing ratios. Similar results can also be obtained for O7 from

Fig. 4b. As a result, it seems also impossible to measure such LNV meson decays practically.

Acknowledgments The work was supported in part by National Center for Theoretical Science, National Science Council (NSC-101-2112-M-007-006-MY3), MoST (MoST-104-2112-M-007-003-MY3) and National Tsing Hua University (104N2724E1).

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/

Web End =http://creativecomm http://creativecommons.org/licenses/by/4.0/

Web End =ons.org/licenses/by/4.0/ ), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

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(ProQuest: ... denotes formulae and/or non-USASCII text omitted; see image)

We study the CP-violation effects from two types of neutrino mass matrices with (i) ......, and (ii) ......, which can be realized by the high-dimensional lepton number violating operators ...... and ......, respectively. In (i), the neutrino mass spectrum is in the normal ordering with the lightest neutrino mass within the range ....... Furthermore, for a given value of ......, there are two solutions for the two Majorana phases ...... and ......, whereas the Dirac phase ...... is arbitrary. For (ii), the parameters of ......, ......, ......, and ...... can be completely determined. We calculate the CP-violating asymmetries in neutrino-antineutrino oscillations for both mass textures of (i) and (ii), which are closely related to the CP-violating Majorana phases.

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Title

CP violations in predictive neutrino mass structures

Author

Geng, Chao-qiang; Huang, Da; Tsai, Lu-hsing

Pages

1-8

Publication year

2015

Publication date

Nov 2015

Publisher

Springer Nature B.V.

ISSN

14346044

e-ISSN

14346052

Source type

Scholarly Journal

Language of publication

English

DOI

https://doi.org/10.1140/epjc/s10052-015-3779-9

ProQuest document ID

1750935970

SIF and Springer-Verlag Berlin Heidelberg 2015

CP violations in predictive neutrino mass structures

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