A full parametrization of the 6\times 6 flavor mixing matrix in the presence of three light or heavy sterile neutrinos

# A full parametrization of the 6×6 flavor mixing matrix in the presence of three light or heavy sterile neutrinos

Zhi-zhong Xing E-mail: xingzz@ihep.ac.cn Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, China
###### Abstract

In addition to three active neutrinos , and , one or more light sterile neutrinos have been conjectured to account for the LSND, MiniBooNE and reactor antineutrino anomalies (at the sub-eV mass scale) or for warm dark matter in the Universe (at the keV mass scale). Heavy Majorana neutrinos at or above the TeV scale have also been assumed in some seesaw models. Such hypothetical particles can weakly mix with active neutrinos, and thus their existence can be detected at low energies. In the (3+3) scenario with three sterile neutrinos we present a full parametrization of the flavor mixing matrix in terms of fifteen rotation angles and fifteen phase angles. We show that this standard parametrization allows us to clearly describe the salient features of some problems in neutrino phenomenology, such as (a) possible contributions of light sterile neutrinos to the tritium beta decay and neutrinoless double-beta decay; (b) leptonic CP violation and deformed unitarity triangles of the flavor mixing matrix of three active neutrinos; (c) a reconstruction of the neutrino mass matrix in the type-(I+II) seesaw mechanism; and (d) flavored and unflavored leptogenesis scenarios in the type-I seesaw mechanism with three heavy Majorana neutrinos.

###### pacs:
PACS number(s): 14.60.Pq, 13.10.+q, 25.30.Pt

## I Introduction

One of the fundamental questions in neutrino physics and cosmology is whether there exist extra species of neutrinos which do not directly participate in the standard weak interactions. Such sterile neutrinos are certainly hypothetical, but their possible existence is either theoretically motivated or experimentally implied. For example,

• heavy Majorana neutrinos at or above the TeV scale are expected in many seesaw models [1], which can not only interpret the small masses of three active neutrinos but also account for the cosmological matter-antimatter asymmetry via the leptogenesis mechanism [2];

• the LSND antineutrino anomaly [3], the MiniBooNE antineutrino anomaly [4] and the reactor antineutrino anomaly [5] can all be explained as the active-sterile antineutrino oscillations in the assumption of two species of sterile antineutrinos whose masses are close to 1 eV [6];

• an analysis of the existing data on the cosmic microwave background (CMB), galaxy clustering and supernovae Ia favors some extra radiation content in the Universe and one or two species of sterile neutrinos at the sub-eV mass scale [7] 111If the bound obtained from the Big Bang nucleosynthesis is taken into account, however, only one species of light sterile neutrinos and antineutrinos is allowed [8].;

• sufficiently long-lived sterile neutrinos in the keV mass range can serve for a good candidate for warm dark matter, whose presence may allow us to solve or soften several problems that we have recently encountered in the dark matter simulations [9] (e.g., to damp the inhomogeneities on small scales by reducing the number of dwarf galaxies or to smooth the cusps in the dark matter halos) 222There are some interesting models which can accommodate sterile neutrinos at either keV [10] or sub-eV [11] mass scales. A model-independent argument is also supporting the conjecture of warm dark matter particles hiding out in the “flavor desert” of the fermion mass spectrum [12]..

No matter how small or how large the mass scale of sterile neutrinos is, they are undetectable unless they mix with three active neutrinos to some extent. The strength of active-sterile neutrino mixing can be described in terms of some rotation angles and phase angles, just like the parametrization of the quark flavor mixing in the standard model [14].

The main purpose of this paper is to present a full parametrization of the flavor mixing matrix in the (3+3) scenario with three sterile neutrinos denoted as , and :

 ⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝νeνμντνxνyνz⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠=U⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝ν1ν2ν3ν4ν5ν6⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠, (1)

where (for ) stand for the mass eigenstates of active and sterile neutrinos. Such a complete parametrization, which has been lacking in the literature [13], is expected to be very useful for the study of neutrino phenomenology at both low and high energy scales. We propose a simple but novel way to establish the connection between active and sterile neutrinos in terms of fifteen mixing angles and fifteen CP-violating phases. It allows us to clearly describe the salient features of some interesting problems, such as (a) possible contributions of light sterile neutrinos to the tritium beta () decay and neutrinoless double-beta () decay; (b) leptonic CP violation and deformed unitarity triangles of the flavor mixing matrix of three active neutrinos; (c) a reconstruction of the neutrino mass matrix in the type-(I+II) seesaw mechanism; and (d) flavored and unflavored leptogenesis scenarios in the type-I seesaw mechanism with three heavy Majorana neutrinos.

## Ii The standard parametrization

The unitary matrix defined in Eq. (1) can be decomposed as

 U=(100U0)(ARSB)(V0001), (2)

in which and stand respectively for the zero and identity matrices, and are the unitary matrices, and , , and are the matrices which satisfy the conditions

 AA†+RR†=BB†+SS†=1, AS†+RB†=A†R+S†B=0, A†A+S†S=B†B+R†R=1, (3)

as a result of the unitarity of . In the limit of , holds and thus there is no correlation between the active sector (described by ) and the sterile sector (characterized by ). In view of Eq. (A7) in Appendix A, we parametrize as follows:

 (V0001) = O23O13O12, (100U0) = O56O46O45, (ARSB) = O36O26O16O35O25O15O34O24O14, (4)

where fifteen two-dimensional rotation matrices (for ) in a six-dimensional complex space have been given in Eqs. (A2)—(A6). To be explicit,

 V0 = ⎛⎜⎝c12c13^s∗12c13^s∗13−^s12c23−c12^s13^s∗23c12c23−^s∗12^s13^s∗23c13^s∗23^s12^s23−c12^s13c23−c12^s23−^s∗12^s13c23c13c23⎞⎟⎠, U0 = ⎛⎜⎝c45c46^s∗45c46^s∗46−^s45c56−c45^s46^s∗56c45c56−^s∗45^s46^s∗56c46^s∗56^s45^s56−c45^s46c56−c45^s56−^s∗45^s46c56c46c56⎞⎟⎠, (5)

in which and with and being the rotation angle and phase angle, respectively. Both and have the standard form as advocated in Ref. [14], and either of them consists of three mixing angles and three CP-violating phases. If the sterile sector is switched off, we are then left with the unitary matrix which describes the flavor mixing of three active neutrinos. If the active sector is switched off, one will arrive at the unitary matrix which purely describes the flavor mixing of three sterile neutrinos. In the type-I seesaw mechanism [1], for example, is essentially equivalent to the unitary transformation used to diagonalize the heavy Majorana neutrino mass matrix and therefore relevant to the leptogenesis mechanism [2].

With the help of Eq. (4) and Eqs. (A2)—(A6), a lengthy but straightforward calculation leads us to the explicit expressions of , , and as follows:

 A = ⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝c14c15c1600−c14c15^s16^s∗26−c14^s15^s∗25c26−^s14^s∗24c25c26c24c25c260−c14c15^s16c26^s∗36+c14^s15^s∗25^s26^s∗36−c14^s15c25^s∗35c36+^s14^s∗24c25^s26^s∗36+^s14^s∗24^s25^s∗35c36−^s14c24^s∗34c35c36−c24c25^s26^s∗36−c24^s25^s∗35c36−^s24^s∗34c35c36c34c35c36⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠, B = ⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝c14c24c3400−c14c24^s∗34^s35−c14^s∗24^s25c35−^s∗14^s15c25c35c15c25c350−c14c24^s∗34c35^s36+c14^s∗24^s25^s∗35^s36−c14^s∗24c25^s26c36+^s∗14^s15c25^s∗35^s36+^s∗14^s15^s∗25^s26c36−^s∗14c15^s16c26c36−c15c25^s∗35^s36−c15^s∗25^s26c36−^s∗15^s16c26c36c16c26c36⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠; (6)

and

 R = ⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝^s∗14c15c16^s∗15c16^s∗16−^s∗14c15^s16^s∗26−^s∗14^s15^s∗25c26+c14^s∗24c25c26−^s∗15^s16^s∗26+c15^s∗25c26c16^s∗26−^s∗14c15^s16c26^s∗36+^s∗14^s15^s∗25^s26^s∗36−^s∗14^s15c25^s∗35c36−c14^s∗24c25^s26^s∗36−c14^s∗24^s25^s∗35c36+c14c24^s∗34c35c36−^s∗15^s16c26^s∗36−c15^s∗25^s26^s∗36+c15c25^s∗35c36c16c26^s∗36⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠, S = (7)

We see that the textures of and are rather similar, so are the textures of and . In fact, the expression of can be obtained from that of with the subscript replacements , , and ; and the expression of can be obtained from that of with the same subscript replacements. Note that the results of and have been obtained in Ref. [15], and here we present the results of and to complete a full parametrization of the flavor mixing matrix .

It proves convenient to define and which describe the flavor mixing phenomena of three active neutrinos and three sterile neutrinos, respectively. Furthermore, links the mass eigenstates to the sterile flavor eigenstates in the chosen basis. We therefore have

 ⎛⎜⎝νeνμντ⎞⎟⎠=V⎛⎜⎝ν1ν2ν3⎞⎟⎠+R⎛⎜⎝ν4ν5ν6⎞⎟⎠, (8)

and

 ⎛⎜⎝νxνyνz⎞⎟⎠=U⎛⎜⎝ν4ν5ν6⎞⎟⎠+ˆS⎛⎜⎝ν1ν2ν3⎞⎟⎠. (9)

Eq. (8) directly leads us to the standard weak charged-current interactions of six neutrinos:

 −Lcc=g√2 ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯(e  μ  τ)L γμ⎡⎢⎣V⎛⎜⎝ν1ν2ν3⎞⎟⎠L+R⎛⎜⎝ν4ν5ν6⎞⎟⎠L⎤⎥⎦W−μ+h.c., (10)

where is just the Maki-Nakagawa-Sakata-Pontecorvo (MNSP) matrix [16] responsible for the active neutrino mixing, and measures the strength of charged-current interactions between and . Because of

 VV†=AA†=1−RR†, U†U=B†B=1−R†R, (11)

we find that both and are not exactly unitary and their non-unitary effects are simply characterized by non-vanishing and .

In view of current observational constraints on sterile neutrinos, we expect that the mixing angles between active and sterile neutrinos are strongly suppressed (at most at the level [6] 333For example, the non-unitarity of or the deviation of from can at most be at the level as constrained by current neutrino oscillation data and precision electroweak data [17].). The smallness of (for and ) allows us to make the following excellent approximations to Eqs. (7) and (8):

 A ≃ 1−⎛⎜ ⎜ ⎜⎝12(s214+s215+s216)00^s14^s∗24+^s15^s∗25+^s16^s∗2612(s224+s225+s226)0^s14^s∗34+^s15^s∗35+^s16^s∗36^s24^s∗34+^s25^s∗35+^s26^s∗3612(s234+s235+s236)⎞⎟ ⎟ ⎟⎠, B ≃ 1−⎛⎜ ⎜ ⎜⎝12(s214+s224+s234)00^s∗14^s15+^s∗24^s25+^s∗34^s3512(s215+s225+s235)0^s∗14^s16+^s∗24^s26+^s∗34^s36^s∗15^s16+^s∗25^s26+^s∗35^s3612(s216+s226+s236)⎞⎟ ⎟ ⎟⎠, (12)

where the terms of have been omitted; and

 R ≃ 0+⎛⎜⎝^s∗14^s∗15^s∗16^s∗24^s∗25^s∗26^s∗34^s∗35^s∗36⎞⎟⎠, S ≃ 0−⎛⎜⎝^s14^s24^s34^s15^s25^s35^s16^s26^s36⎞⎟⎠, (13)

where the terms of have been omitted. It turns out that holds in the same approximation.

Note that the unitary matrix can be used to describe not only the flavor mixing between active and sterile neutrinos but also the flavor mixing between ordinary and extra quarks. Note also that it is straightforward to obtain the (3+1) flavor mixing scenario from Eqs. (6) and (7) by switching off the mixing angles and (for and ), or the (3+2) flavor mixing scenario by turning off the mixing angles (for ).

## Iii Some applications

To illustrate the usefulness of our parametrization of the flavor mixing matrix , let us briefly discuss its four simple but instructive applications in neutrino phenomenology.

### iii.1 The effective masses of β and 0ν2β decays

One or two light sterile neutrinos at the sub-eV mass scale have been hypothesized for a quite long time to interpret the LSND antineutrino anomaly [3] and the subsequent MiniBooNE antineutrino puzzle [4]. In general, however, it seems more natural to assume the number of sterile neutrino species to be equal to that of active neutrino species [18] 444In order to avoid any severe conflict between such a (3+3) scenario and the standard CDM cosmology, it is perhaps necessary to either loosen the mass hierarchy of three sterile neutrinos (i.e., not all of them are of eV) or refer to some nonstandard models of cosmology [7]., such that even possible warm dark matter in the form of one or two species of keV sterile neutrinos could be taken into account.

For simplicity and illustration, we are only concerned about the effective masses of the tritium beta () decay and the neutrinoless double-beta () decay in the (3+3) neutrino mixing scenario. The former is

 ⟨m⟩′e ≡ [6∑i=1m2i|Vei|2]1/2=√⟨m⟩2ec214c215c216+m24s214c215c216+m25s215c216+m26s216, (14)

where is the standard contribution from three active neutrinos. We see that always holds. The effective mass of the decay is

 ⟨m⟩′ee ≡ 6∑i=1miV2ei=⟨m⟩ee(c14c15c16)2+m4(^s∗14c15c16)2+m5(^s∗15c16)2+m6(^s∗16)2 (15)

with being the standard contribution from three active neutrinos. It is difficult to say about the relative magnitudes of and , because the CP-violating phases may give rise to more or less cancelations of different terms in them. In particular, even [19] or [20] is not impossible. If both and can be determined or constrained in the future experiments, a comparison between them might be able to probe the existence of light sterile neutrinos [21].

### iii.2 Deformed unitarity triangles and CP violation

Switching off three sterile neutrinos, one may describe flavor mixing and CP violation of three active neutrinos in terms of six unitarity triangles in the complex plane [22]. Three of them, defined by the orthogonality conditions

 △e: Vμ1V∗τ1+Vμ2V∗τ2+Vμ3V∗τ3=0, △μ: Vτ1V∗e1+Vτ2V∗e2+Vτ3V∗e3=0, △τ: Ve1V∗μ1+Ve2V∗μ2+Ve3V∗μ3=0, (16)

are illustrated in FIG. 1 (left panel). The area of each triangle is equal to , where is the Jarlskog parameter given in Eq. (B2) and measures the strength of leptonic CP-violating effects in , and oscillations. Now let us turn on the contributions of three sterile neutrinos to flavor mixing and CP violation. Then Eq. (16) approximates to

 △′e: Vμ1V∗τ1+Vμ2V∗τ2+Vμ3V∗τ3≃−Z∗, △′μ: Vτ1V∗e1+Vτ2V∗e2+Vτ3V∗e3≃−Y, △′τ: Ve1V∗μ1+Ve2V∗μ2+Ve3V∗μ3≃−X∗, (17)

where , and . These deformed unitarity triangles are also illustrated in FIG. 1 (right panel). The small differences of their areas from just signify the new CP-violating effects.

Let us take a look at the CP-violating asymmetries between and oscillations, defined as . With the help of Eq. (B5), we explicitly obtain

 Aμe ≃ −4(J0+c12s12c23ImX)sinΔm221L2E, Aeτ ≃ −4(J0+c12s12s23ImY)sinΔm221L2E, Aμτ ≃ +4[J0+c12s12c23s23(s23ImX+c23ImY)]sinΔm221L2E+4c23s23ImZsinΔm232L2E, (18)

where , and . We see that and are related to the deformed unitarity triangles and , respectively. In comparison, has something to do with . It is therefore possible to determine three new CP-violating terms , and by measuring the CP-violating effects in neutrino oscillations. Note that three CP-violating asymmetries in Eq. (18) satisfy the correlation

 Aμτ+(s223Aμe+c223Aeτ)≃4c23s23ImZsinΔm232L2E. (19)

When holds, both and are suppressed such that becomes a pure measure of the non-unitary CP-violating parameter . This interesting possibility, together with terrestrial matter effects, has been discussed before (e.g., Refs. [15] and [26]).

### iii.3 Reconstruction of the 6×6 neutrino mass matrix

The type-(I+II) seesaw mechanism [27] is a good example to illustrate the flavor mixing between three active neutrinos and three heavy Majorana neutrinos. In this mechanism the mass term of six neutrinos is usually written as

 −Lmass=12 ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯(νL NcR) (MLMDMTDMR)(νcLNR)+h.c., (20)

where and represent the column vectors of three left-handed neutrinos and three right-handed neutrinos, respectively. The overall neutrino mass matrix in Eq. (20) can be diagonalized by a unitary transformation:

 U†(MLMDMTDMR)U∗=(ˆMν00ˆMN), (21)

where is already given in Eq. (2), and with or (for ) being the physical masses of light or heavy Majorana neutrinos. The standard weak charged-current interactions of six neutrinos are given by Eq. (10) with , and in the basis of mass eigenstates. With the help of Eq. (2),

 ML = VˆMνVT+RˆMNRT≃V0ˆMνVT0+RˆMNRT, MD = VˆMνˆST+RˆMNUT≃RˆMNUT0, MR = ˆSˆMνˆST+UˆMNUT≃U0ˆMNUT0, (22)

where , and have been defined before. The approximations made on the right-hand side of Eq. (22) follow the spirit that only the leading terms of , and are kept. It is then possible to reconstruct these neutrino mass matrices, at least in principle, in terms of neutrino masses and flavor mixing parameters [28].

Given the basis where is diagonal, real and positive, Eq. (22) implies that together with is a good approximation. Note that such a flavor basis is often chosen in the study of leptogenesis, because the decays of (for ) need to be calculated. It is also much easier to reconstruct and in this special basis. For example,

 MD≃RˆMN≃⎛⎜⎝M1^s∗14M2^s∗15M3^s∗16M1^s∗24M2^s∗25M3^s∗26M1^s∗34M2^s∗35M3^s∗36⎞⎟⎠; (23)

and six independent matrix elements of can similarly be obtained:

 (ML)ee ≃ m1(c12c13)2+m2(^s∗12c13)2+m3(^s∗13)2+M1(^s∗14)2+M2(^s∗15)2+M3(^s∗16)2, (ML)eμ ≃ −m1c12c13(^s12c23+c12^s13^s∗23)+m2^s∗12c13(c12c23−^s∗12^s13^s∗23)+m3c13^s∗13^s∗23 +M1^s∗14^s∗24+M2^s∗15^s∗25+M3^s∗16^s∗26, (ML)eτ ≃ m1c12c13(^s12^s23−c12^s13c23)−m2^s∗12c13(c12^s23+^s∗12^s13c23)+m3c13^s∗13c23 +M1^s∗14^s∗34+M2^s∗15^s∗35+M3^s∗16^s∗36, (ML)μμ ≃ m1(^s12c23+c12^s13^s∗23)2+m2(c12c23−^s∗12^s13^s∗23)2+m3(c13^s∗23)2 +M1(^