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University of Wisconsin - Madison

MADPH-99-1096 ASITP-99-05 AMES-HET-99-01

January 1999

NEUTRINO MIXING, CP=T VIOLATION AND TEXTURES IN FOUR-NEUTRINO MODELS

2 3

V. Barger , Yuan-Ben Dai , K. Whisnant , and Bing-Lin Young Department of Physics, University of Wisconsin, Madison, WI 53706, USA Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100080, China Department of Physics and Astronomy, Iowa State University, Ames, IA 50011, USA

1 2 3 3 1

Abstract

We examine the prospects for determining the neutrino mixing matrix and for observing CP and T violation in neutrino oscillations in four-neutrino models. We focus on a general class of four-neutrino models with two pairs of nearly degenerate mass eigenstates separated by approximately 1 eV, which can describe the solar, atmospheric and LSND neutrino data. We present a general parametrization of these models and discuss in detail the determination of the mixing parameters and the mass matrix texture from current and future neutrino data in the case where e and each mix primarily with one other neutrino. We nd that measurable CP=T violation in long-baseline experiments, with amplitude at the level of the LSND signal, is possible given current experimental constraints. Also, additional oscillation e ects in shortand long-baseline experiments may be measurable in many cases. We point out that, given separate scales for the mass-squared di erences of the solar and atmospheric oscillations, observable CP=T violation e ects in neutrino oscillations signals the existence of a sterile neutrino. We examine several textures of the neutrino mass matrix and determine which textures can have measurable CP=T violation in neutrino oscillations in long-baseline experiments. We also brie y discuss some possible origins of the neutrino mass terms in straightforward extensions of the Standard Model.

I. INTRODUCTION

Our view of the neutrino sector of the Standard Model has recently undergone a revolutionary change. Observations of solar neutrinos 1{5], atmospheric neutrinos 6{8], and accelerator neutrinos 9] all indicate deviations from their predicted values in the Standard Model with massless neutrinos. In each case the observation can be understood in terms of neutrino oscillations which in turn requires nondegenerate neutrino masses. The accelerator evidence for oscillations is least secure, with preliminary data from the KARMEN Collaboration 10] excluding some regions of oscillation parameters preferred by the LSND data 9]. Since the solar, atmospheric, and LSND neutrino experiments have di erent L=E (the ratio of oscillation distance to neutrino energy), di erent orders of magnitude of neutrino masssquared di erences m are required to properly describe all features of the data 11]. This need for three small but distinct mass-squared di erences naturally leads to the consideration of more than three light neutrino avors. Any additional light neutrino must be sterile, i.e., without Standard Model gauge interactions, to be consistent with the well-established LEP measurements of Z ! 12]. From quite general arguments it has been shown 13,14] that a neutrino spectrum with two pairs of nearly degenerate mass eigenstates, separated by a gap of order 1 eV, is required to satisfy all of the constraints from solar, atmospheric, accelerator, and reactor data. Sterile neutrinos (which we we denote as s ) have long been considered as an option for neutrino oscillations 15,16]. More recently a number of models have been proposed that utilize one or more sterile neutrinos to describe the existing neutrino data 14,17{20] or to explain r-process nucleosynthesis 21]. However, if sterile neutrinos mix with active avor neutrinos they may be stringently constrained by Big Bang nucleosynthesis (BBN). In standard BBN phenomenology, the mass-squared di erence m and the mixing angle between a sterile and active neutrino must satisfy the bound

2 2

m sin 2 < 10? eV ;

2 2 7 2

(1)

to avoid thermal overpopulation of the \extra", sterile neutrino species 22]. The restriction in Eq. (1) would appear to rule out all sterile-active mixing except for small-angle MSW or vacuum mixing of solar neutrinos. However, some recent estimates of N using higher inferred abundance of He yield a considerably weaker bound than that given in Eq. (1) 23]. Thus BBN may still allow sizeable mixing between sterile and active neutrinos, so models with both small and large mixings with sterile neutrinos can be considered. In this paper we examine the phenomenological consequences of four-neutrino models in which there are two pairs of neutrinos with nearly degenerate mass eigenstates separated by about 1 eV, where the mass separations within the pairs are several orders of magnitude smaller. We begin with a general parametrization of the four-neutrino mixing matrix, and review the current experimental constraints. We then discuss the simple situation where e mixes dominantly with s or in solar neutrino oscillations and mixes dominantly with a fourth neutrino ( or s ) in atmospheric neutrino oscillations. This situation, which we refer to as the dominant mixing scheme, has been shown to t the existing data reasonably well. For dominant mixing we nd that the neutrino mixing matrix can be e ectively analyzed in terms of 2 2 blocks, where the diagonal blocks can be approximated by simple twoneutrino rotations and the o -diagonal blocks are small but non-vanishing. We then study

4

2

the relationship between the phenomenology of neutrino oscillations with CP violation and the texture of the neutrino mass matrix in models where the two lightest states are much lighter than the two heaviest states. Since the neutrino mass matrix can in general be complex, and therefore lead to a mixing matrix with complex elements, CP violation can naturally arise in neutrino oscillations. The pertinent question is the size of the violation and how to observe it. We nd that if CP violation exists, its size may be measurable, and has approximately the same amplitude as indicated by the LSND experiment. In many cases there are small amplitude e ! oscillations that may be measurable in either short- or long-baseline experiments. Furthermore, some also have small amplitude ! oscillations in short-baseline experiments. We discuss how oscillation measurements in solar, atmospheric, short- and long-baseline neutrino experiments can, in some cases, determine all but one of the four-neutrino mixing matrix parameters accessible to oscillation measurements. We also discuss the minimal Higgs boson spectrum needed to obtain the di erent types of four-neutrino mass matrices, and their consequences for CP violation. This paper is organized as follows. In Sec. II we present our parametrization for the four-neutrino mixing matrix and expressions for the oscillation probabilities in the case of two pairs of nearly degenerate mass eigenstates separated by about 1 eV. We discuss the ways in which CP violation, if it exists, may be observed, and the number of observable CP violation parameters. In Sec. III we summarize the current constraints on the four-neutrino mixing matrix and discuss in detail the implications of the dominant mixing scheme. We investigate the CP violation e ects for several mass matrix textures in Sec. IV. In Sec. V we brie y discuss some of the consequences of neutrino mass for non-oscillation experiments, such as rare decays and charged lepton electric dipole moments, and we emphasize the importance in searching for these rare events, which can reveal new physics e ects other than neutrino masses. In Sec. VI we summarize our results. Finally, in Appendix A we review the number of independent parameters in the mixing matrix of Majorana neutrinos, in Appendix B we discuss the modest extensions of the Standard Model Higgs sector that allow us to obtain the mass matrix textures, and in Appendix C we determine the neutrino mass spectrum and mixing matrix for a particular neutrino mass matrix with CP violation.

II. OSCILLATION PROBABILITIES A. General Formalism

We work in the basis where the charged lepton mass matrix is diagonal. The most general neutrino mass matrix M is Majorana in nature, and may be diagonalized by a complex orthogonal transformation into a real diagonal matrix

MD = U T MU ;

(2)

by a unitary matrix U , which is generally obtained from the Hermitian matrix M yM = M M by MD MD = U yM yMU 24]. Some general properties of Majorana neutrino mass matrices are discussed in more detail in Appendix A. For the four-neutrino case, labeling the avor eigenstates by x; e; ; y and the mass eigenstates by ; ; ; , we may write

0 1 2 3

3

0 B B B @

x C B e C=UB C B y

1 A

0 1 C C A @ C:

0 1 2 3

(3)

In this paper we will examine the two cases most often considered in the recent literature: one of x and y is and the other is sterile ( s), or both x and y are sterile. Explicitly, the matrix M may be written in the avor basis as 0M M M M 1 B Mxx Mxe Mx Mxy C M = B Mxe Mee Me M ey C : B C (4) @ x e yA Mxy Mey M y Myy The 4 4 unitary matrix U may be parametrized by 6 rotation angles and 6 phases, and can be conveniently represented by 25] U =R R R R R R ; (5) where 0 c s 0 01 B s 0 0C C (6) R = B ?0 c0 1 0 C ; B A @ 0 0 01 with cjk cos jk ; sjk sin jk ei jk ; (7) and the other Rjk are de ned similarly for rotations in the j {k plane. The explicit form for the 4 4 unitary matrix is 0 c c c 1 c c s c s s B C B C B ?c c s s B C c s C ?s s s ?c s s s B C B ?c c s s C +c c s ?c s s s B C B ?c c s C B C +c c c B C B C B B ?c c c s s ?c c s s s ?c s s s c c s C C B C B +c s s s s +s s s s s ?c s s s C B C B C B ?c c c s C ?c c s s +c c c B C: U = B +c s s s C (8) B C ?c c s s B C B +c s s C ?c c s B C B C B B ?c c c c s ?c c c s s ?c c s s c c c C C B C B C B +c c s s s +c s s s s ?c c s s C B C B +c c s s C B C +c s s s ?c c s B C B +c c s s C ?c c c s B C B ?s s s C @ A +c s s

23 13 03 12 02 01 01 01 01 01 01 01 02 03 02 03 01 03 02 03 01 02 03 13 02 01 03 13 02 03 13 03 13 01 13 02 12 12 13 01 13 01 02 12 01 12 13 02 13 12 01 02 13 03 23 02 13 01 03 23 13 02 03 23 03 13 23 01 02 12 13 23 01 12 23 02 01 02 12 13 23 12 23 01 02 02 12 13 23 02 12 23 12 01 13 23 23 01 12 01 12 13 23 01 23 12 01 02 13 23 03 02 13 23 01 03 13 23 02 03 03 13 23 01 23 02 12 13 01 12 02 23 23 01 02 12 13 12 01 02 23 01 12 23 13 01 12 23 02 23 12 13 02 12 23 12 23 01 13 01 12 23

4

We will label the matrix elements of U by U j , where Greek indices denote avor eigenstate labels ( = x; e; ; y) and Latin indices denote mass eigenstate labels (j = 0; 1; 2; 3). With the knowledge of the mass eigenvalues and mixing matrix elements, one can invert Eq. (2) to obtain the neutrino mass matrix elements X (9) M = U j U j mj :

3

j =0

The vacuum neutrino avor oscillation probabilities, for an initially produced nally detected , can be written i Xh P ( ! ) = ? 4 Re(W jk ) sin kj ? 2 Im(W jk ) sin 2 kj ;

2

to a (10) (11) (12)

j<k

where

W jk

kj

U j U kU j U k ; mkj L=(4E ) ;

2

mkj mk ? mj ;

2 2 2

L is the oscillation distance, and E is the neutrino energy. The quantities W jk 26], are related to the Jarlskog invariants 27] J jk

which satisfy the identity Im(W jk ) ; (13) (14) (15) (16) (17)

J jk = ?J jk ;

obtained by the interchange U $ U in Eq. (11). Also,

J jk = J kj = ?J jk = ?J kj :

We can also de ne the real part of W jk as

Y jk Re(W jk ) ;

which is invariant under interchange of the upper or lower indices:

Y jk = Y kj = Y jk = Y kj :

Another useful property of the W jk is that the sum over any of the indices reduces them to a real positive quantity, e.g., X jk X W = jU j j jk = Y jk ; (18) X jk (19) J = 0:

2

Equations (10), (14), (15) and (17) imply 5

P( ! ) = P( ! );

(20)

which is a statement of CPT invariance. Equation (10) and (15) imply that nonzero J jk can give CP or T violation

P ( ! ) 6= P ( ! ) = P ( ! ) : (21) From Eq. (21) we can de ne the CP -violation quantity P = P( ! ) ? P( ! ): (22) In four-neutrino oscillations there are only three independent P , and, correspondingly, only three of the six phases in U can be measured in neutrino oscillations (for a discussion, see Appendix A). Thus six angles and three phases can in principle be measured in neutrino oscillations, which is the same as in the Dirac neutrino case. Therefore, as far as neutrino oscillations are concerned, our results apply equally to Dirac neutrinos. The three remaining independent phases in U enter into the mass matrix elements and processes such as neutrinoless double beta decay.

B. Model with two nearly degenerate pairs of neutrinos

For a four-neutrino model to describe the solar, atmospheric, LSND data and also satisfy all other accelerator and reactor limits, it must have two pairs of nearly degenerate mass mLSND m . We will also matm m eigenstates 13,14]; e.g., msun m assume without loss of generality that 0 < m ; m < m < m . An alternative scenario with the roles of m and m reversed gives the same results as far as oscillations are concerned, although the implications for the mass matrix, double beta decay and cosmology may di er; this alternate possibility will be brie y discussed in Sec. IV.E. Also note that if the solar oscillations are driven by the MSW e ect 28], we must require m > m ; for vacuum oscillations, m < m is also possible. Given this hierarchy of the m , the oscillation probabilities for 6= may be written approximately as

2 2 01 2 2 32 2 2 21 2 01 2 32 0 1 2 3 0 1 0 1 2

P ( ! ) ' ALSND sin LSND + Aatm sin atm + Batm sin 2 atm +Asun sin sun + Bsun sin 2 sun ; 6= ; (23) and for the diagonal channels P ( ! ) ' 1 ? ALSND sin LSND ? Aatm sin atm ? Asun sin sun ; (24) where scale mscaleL=E , Ascale is the usual CP conserving oscillation amplitude for ! at a given oscillation scale, Bscale is the CP violation parameter at a given scale, and the scale label is sun for the solar neutrino scale, atm for atmospheric and long-baseline scales, and LSND for accelerator and short-baseline scales. Note that the CP -violating terms have a di erent dependence on L=E from the CP -conserving terms, which could in principle be distinguished by measurements at di erent L=E 29]. In Eqs. (23) and (24) we have used the approximation ' ' ' ' LSND.

2 2 2 2 2 2 1 4 2 31 30 21 20

6

The oscillation amplitudes are given by 6= ; (25) ALSND = 4jU U + U U j = 4jU U + U U j ; ALSND = 4(jU j + jU j )(1 ? jU j ? jU j ) ; = 4(jU j + jU j )(1 ? jU j ? jU j ) ; (26) Aatm = ?4 Re(U U U U ) ; 6= ; (27) Aatm = 4jU j jU j ; (28) Asun = ?4 Re(U U U U ) ; 6= ; (29) Asun = 4jU j jU j ; (30) where the second equality in Eqs. (25) and (26) follows from the unitarity of U . We note that the form of the short-baseline oscillation amplitudes in Eqs. (25) and (26) are di erent from the cases of long-baseline, Eqs. (27) and (28), and solar, Eqs. (29) and (30). The di erence is due to the fact that the short-baseline oscillations arise from four mass-squared di erences ( m ' m ' m ' m ), while the long-baseline and solar oscillations arise from only one mass-squared di erence ( m and m , respectively). Probability conservation implies Ascale = P 6 Ascale, which can easily be shown using the unitarity of U . The CP violation parameters are (31) Batm = ?2 Im(U U U U ) ; (32) Bsun = 2 Im(U U U U ) : Since Bj = 0, there is no CP violation in diagonal channels. The absence of BLSND in Eq. (23) shows that no observable CP violation is present for the leading oscillation 30], and CP violation may only be seen in experiments that probe non-leading scales, matm or msun. For short-baseline experiments where only the leading oscillation argument LSND has had a chance to develop, the o -diagonal vacuum oscillation probabilities are 6= ; (33) P ( ! ) ' ALSND sin LSND ; P ( ! ) ' 1 ? ALSND sin LSND : (34) For larger L=E (such as in atmospheric and long-baseline experiments), where the secondary oscillation has had time to develop, the vacuum oscillation probabilities are 1 6= ; (35) P ( ! ) ' 2 ALSND + Aatm sin atm + Batm sin 2 atm ; P ( ! ) ' 1 ? 1 ALSND ? Aatm sin atm : (36) 2

2 2 3 3 2 0 0 1 1 2 2 2 3 2 2 2 3 2 0 2 1 2 0 2 1 2 2 3 2 3 2 2 3 2 0 1 0 1 0 2 1 2 2 20 2 30 2 21 2 31 2 32 2 01 = 2 3 2 3 0 1 0 1 2 2 2 2 2 2

Here we have assumed that the leading oscillation has averaged, i.e., sin LSND ! . Finally, at the solar distance scale, when all oscillation e ects have developed, the vacuum oscillation probabilities are 6= ; (37) P ( ! ) ' 1 (ALSND + Aatm) + Asun sin sun + Bsun sin 2 sun ; 2 P ( ! ) ' 1 ? 1 (ALSND + Aatm) ? Asun sin sun ; (38) 2

2 1 2 2 2

7

where sin atm has been averaged to and sin 2 atm has been averaged to 0; the CP violation at the matm scale is washed out. CP violation is possible only in the o -diagonal channels, as noted before, and the solar neutrino e survival measurement cannot be used to observed CP violation. More generally, in any model for which the oscillation scales are well-separated and L=E is only large enough to probe the largest oscillation scale, CP -violating e ects in neutrino oscillations will be unobservable (strictly speaking, they are suppressed to order 1, where is the oscillation argument for the second-largest oscillation scale) 30]. The CP violating e ects become observable when L=E is large enough to probe both the largest and second-largest oscillation scales. For a four-neutrino model with di erent oscillation scales to describe the solar, atmospheric and LSND data, this means that CP violation can only be detected in experiments with L=E at least as large as those found in atmospheric and long-baseline experiments. As a corollary, in three-neutrino models with two oscillation scales describing only the solar and atmospheric data, CP violation has the potential to be observable only in experiments with L=E comparable to or larger than the solar experiments. However, a measurement of o -diagonal oscillation probabilities is required to see CP violation, and that is not possible in solar neutrino experiments. Hence, if the solar neutrino oscillation scale is well-established, the observation of a CP -violation e ect in long-baseline experiments could imply that there are at least three separate neutrino mass-squared di erence scales, and thus more than three neutrinos.

2 2 1 2

III. DETERMINING THE OSCILLATION PARAMETERS

In this section we rst derive some general constraints imposed by current data on the neutrino mixing matrix for four-neutrino models favored by the data, i.e., with two pairs m . We then determine the form m of nearly degenerate masses satisfying m of the mixing matrix under the assumption that e and x are mostly a mixture of and and provide the dominant solar neutrino oscillation, and and y are mostly a mixture of and and provide the dominant atmospheric neutrino oscillation, which is the form of most explicit models in the literature. Then we discuss the measurements needed to determine the parameters in the neutrino mixing matrix. The results of this section apply equally to the case where m < m < m ; m . The more general case where e and have large mixing with more than one other neutrino is brie y discussed in Sec. IV.E.

2 01 2 32 2 21 0 1 2 3 2 3 0 1

A. Solar e !

x

and atmospheric

!

y

The avor eigenstates are related to the mass eigenstates by Eq. (3). Using the formulae in Sec. II, the amplitudes for short-baseline oscillation (such as LSND, reactors, and other past accelerator oscillation searches) are

Ae LSND = 4jUe U + Ue U j = 4jUe U + Ue U j ; ALSND = 4(jU j + jU j )(1 ? jU j ? jU j ) = 4(jU j + jU j )(1 ? jU j ? jU j ) ;

2 2 3 3 2 0 0 1 1 2 2 2 3 2 2 2 3 2 0 2 1 2 0 2 1 2

(39) (40)

8

Aee = 4(jUe j + jUe j )(1 ? jUe j ? jUe j ) LSND = 4(jUe j + jUe j )(1 ? jUe j ? jUe j ) ;

2 2 3 2 2 2 3 2 0 2 1 2 0 2 1 2

(41)

where the second equalities in each case result from the unitarity of U . For atmospheric and long-baseline oscillation, the amplitudes are

Aatm = 4jU j jU j ; Ae = ?4 Re(Ue Ue U U ) ; atm e Batm = ?2 Im(Ue Ue U U ) ; Aey = ?4 Re(Ue Ue Uy Uy ) ; atm ey Batm = ?2 Im(Ue Ue Uy Uy ) ; y Aatm = ?4 Re(U U Uy Uy ) ; y Batm = ?2 Im(U U Uy Uy ) ;

2 2 3 2 2 3 2 3 2 3 2 3 2 3 2 3 2 3 2 3 2 3 2 3 2 3 2 3

(42) (43) (44) (45) (46) (47) (48) (49) at the atmospheric scale (50)

and in solar experiments

Aee = 4jUe j jUe j : sun

0 2 1 2

The atmospheric neutrino experiments favor large mixing of 14,31]

Aatm > 0:8 ;

at 90% C.L.; then Eq. (42) and unitarity imply

jU j + jU j > 0:894 ;

2 2 3 2

jU j + jU j < 0:106 :

0 2 1 2

(51) (52)

Also, the Bugey reactor constraint 32] gives

Aee < 0:06 ; LSND

over the indicated range for mLSND; then Eq. (41) implies

2

jUe j + jUe j < 0:016 ;

2 2 3 2

jUe j + jUe j > 0:984 :

0 2 1 2 2

(53) (54) (55)

Finally, the oscillation interpretation of the LSND results 9] gives

Ae LSND

;

where is experimentally constrained to the range 9] 0:05 < < 0:20 ;

2 2 3 0 1

where the exact value depends on mLSND. Hence, the atmospheric and Bugey results imply that jUe j, jUe j, jU j, and jU j are all approximately of order or smaller. We note that given these constraints the size of ALSND must also be small, in agreement with the CDHS bound on disappearance 33]. 9

If we assume that it is only y that mixes appreciably with in the atmospheric experiments and only x that mixes appreciably with e in the solar experiments, then jUx j, jUx j, jUy j, and jUy j must also be small, i.e., of order or less. The mixing matrix can therefore be seen to have the form ! U U ; U= U U (56) where the Uj (j = 1; 2; 3; 4) are 2 2 matrices and the elements of U and U are at most of order in size. The matrix U is approximately the 2 2 maximal mixing matrix p (i.e., all elements have approximate magnitude 1= 2) that describes atmospheric ! y oscillations, but U , which is approximately unitary by itself and which primarily describes the mixing in the solar neutrino sector, may have large (for vacuum oscillations) or small (for MSW oscillations) mixing. The form of U in Eq. (56), with U and U , implies js j; js j; js j; js j ; (57) in the general parametrization of Eq. (8). After dropping terms second order in and smaller, U takes the form 0 1 s s C c s B B C B C B ?s c s s C B C B C B B ?c (s s + c s ) ?s (s s + c s ) c s C C B C B C: U ' B +s (s s + c s ) ?c (s s + c s ) (58) C B C B C B C B B c (s s ? c s ) s (s s ? c s ) ?s c C C B C B ?s (s s ? c s ) +c (s s ? c s ) C @ A

2 3 0 1 1 2 4 3 2 3 4 1 2 3 02 03 12 13 01 01 02 03 01 01 12 13 01 23 03 23 02 01 23 03 23 02 23 23 01 23 13 23 12 01 23 13 23 12 01 23 02 23 03 01 23 02 23 03 23 23 01 23 12 23 13 01 23 12 23 13

This matrix provides a general parametrization of the four-neutrino mixing in models where e mixes primarily with x at the solar mass-squared di erence scale, and mixes primarily with y at the atmospheric mass-squared di erence scale. Unitarity of U is satis ed to the order of . Despite the fact that the expansion of the matrix elements of U in Eq. (58) is to the rst order of , it still allows us, as shown below, to extract all of the interesting oscillation and CP violation e ects, which are second order in . Care must be taken when the leading order result cancels, and sometimes it is helpful to use the unitarity of U to derive an alternate y expression that gives the correct leading order answer, e.g., for ALSND , the rst expression in Eq. (25) gives zero when the form of U in Eq. (58) is used, but the second expression gives a nite (and correct to leading order) result. The o -diagonal oscillation amplitudes for the leading oscillation are Ae (59) LSND = 4js c + s s j ; ey (60) ALSND = 4js s ? s c j ; x ALSND = 4js c + s s j ; (61) y ex ALSND = ALSND = O( ) : (62)

12 23 13 23 2 12 23 02 23 4 13 23 2 03 23 2

10

For atmospheric and long-baseline experiments the oscillation amplitudes are

Ae = ?Aey = ?4c Re(s s s ) ; atm atm e = ?B ey = ?2c Im(s s s ) ; Batm atm ex ; B ex = O( ) ; Aatm atm y Aatm ' Aatm = sin 2 ; x Aatm = ?4c Re(s s s ) ; x Batm = 2c Im(s s s ) ; y Batm = ?2c Im (s s + s s )s ] ;

23 12 13 23 23 12 13 23 4 2 23 23 02 03 23 23 02 03 23 02 03 23 12 13 23 23 2 2

(63) (64) (65) (66) (67) (68) (69)

where is de ned in Eq. (7). All oscillation amplitudes at the LSND scale and all oscillation y amplitudes (including the CP violation amplitudes) other than Aatm at the atmospheric y y scale are at most of order . Since Aatm is large, Batm, which is of order , may be hard to measure since it involves taking the di erence of two nearly equal large numbers. At the solar scale, we have

Aee ' Aex = sin 2 : sun sun

2 01 01 02 03 12 13 23

(70)

There are 6 mixing angles and 3 independent phases that in principle may be measured in neutrino oscillations. In all, there are eight independent parameters involved in the observables in Eqs. (59){(70), which are the six mixing angles , , , , , , and the two phases

0 1 03 13

? ?

02 12

? ; ? :

23 23

(71) (72)

These eight parameters could in principle be determined by measurements of the eight obx x e x servables Aee , Aatm, Ae , Ae , Batm, ALSND, Aatm, and Batm. Therefore if x = , LSND atm sun then all eight of these parameters could in principle be determined from the solar, atmospheric, short- and long-baseline experiments. This emphasizes the need for both shortand long-baseline measurements of all active oscillation channels, since the oscillation amplitudes involve di erent combinations of the parameters at short and long baselines. If x is sterile, the three parameters , and might be di cult to determine since they would involve the disappearance ! s that is at most of order in magnitude. If y = , ey the additional observables Aey , Aey , and Batm can provide a consistency check on the LSND atm parameters , , and . We note that from the above results that many of the CP -violating amplitudes can be the same order of magnitude as the corresponding CP -conserving amplitudes, and hence potentially observable in high-statistics long-baseline experiments. The CP violation parameters x e Batm and Batm could be determined in vacuum by measuring probability di erences Pe and P x, where

02 03 0 2 12 13 1

P P( ! ) ? P( ! ); (73) in long-baseline experiments, or the probability di erences Pe and P x , where P , de ned in Eq. (22), measures explicit T -violation. In a vacuum,

11

P = P = 2Batm sin 2

Alternatively, one could measure CP P( ACP = P ( or T asymmetries P( AT = P (

atm :

(74) (75)

asymmetries ! ) ? P( ! ) ; ! ) + P( ! )

! ) ? P( ! ) : ! ) + P( ! )

(76)

In vacuum, CPT invariance insures that P = P and AT = ACP . However, matter e ects could induce a nonzero P or ACP even in the absence of CP violation 18,34]. Since the matter e ects in long-baseline experiments for P ( ! ) and P ( ! ) are the same, the quantities Pe and AT , which can only be nonzero if there is explicit CP e or T violation, may be preferable 18]. There remains a third independent phase that could have consequences for neutrino oscillations, but will in practice be di cult to measure. This phase, which could be taken as de ned in Eq. (5), could be determined from CP violation in e $ or y at the msun scale, but this e ect would require the measurement of an o -diagonal channel at the solar scale. Therefore it appears that a complete determination of the four-neutrino mixing matrix is not possible with conventional oscillation experiments. Table I lists all parameters that appear to be accessible to observation together with the principal observables that determine these parameters.

01 2

B. More general mixing scenarios

In general, both solar e and atmospheric could oscillate into mixtures of x and y . In this event and in Eq. (8) are not necessarily small. If one of x and y is the tau neutrino and the other sterile, there are several possible ways that the existence of such mixing could be determined 35]. Also, vacuum CP -violation e ects involving e will still be no larger than order (due to the smallness of Ue and Ue ), but there are potentially large CP -violation e ects in long-baseline - y oscillations (as large as allowed by the unitarity of U ) 18].

02 03 2 2 3

IV. CP VIOLATION AND NEUTRINO MASS TEXTURES

In this section we study the relationship between the neutrino mass texture and the possibility for observable CP violation (and, equivalently, T violation) in neutrino oscillations in four-neutrino models. We will consider models where one of x and y is sterile and the other is (such as in Refs. 14], 19], 20], and 36]), and also models where both are sterile 37], which are two possible extensions of the Standard Model neutrinos. Note that in all earlier studies the mass matrices were taken to be real and no CP violation was possible. In Appendix B we discuss straightforward extensions of the Standard Model for the two 12

cases and show explicitly how their neutrino mass matrices can arise. In Secs. IV.A{IV.D we assume that q (77) m ' m m ' m ' mLSND ; i.e., the lighter pair of nearly-degenerate mass eigenstates are much lighter than the heavier pair, also nearly degenerate, which is the structure of most explicit four-neutrino models in the literature. In Sec. IV.E we brie y discuss models with other mass hierarchies. From Eqs. (9) and (58) the neutrino mass matrix elements involving and y are, to leading order in , M ' Myy ' m(c + s ) ; (78) M y ' im sin2 sin ; (79) Me ' m(s c + s s ) ; (80) Mey ' m(s c ? s s ) ; (81) Mx ' m(s c + s s ) ; (82) Mxy ' m(s c ? s s ) ; (83) where we have used the relative sizes of the U j and mass eigenvalues, and the fact that to leading order in , m ' m m. Note that the sjk , de ned in Eq. (7), may be complex. For the mass matrix elements Mxx, Mxe, and Mee , all four terms in Eq. (9) are small and may be of similar size (the rst two are suppressed by the small values of m and m , the last two by mixing angles of size ); their values depend on the exact structure in the solar sector, which we do not specify here. More precise solar neutrino measurements would help to determine their values. The three phases which enter in the expressions for the mass matrix elements given above are and 0 ? + ; (84) 0 ? + : (85) Only the phases and , which can be measured in oscillation experiments, and 0 , which appears in the expressions for the mass matrix elements, are independent. The two phases = ( 0 ? )=2 and 0 = 0 + ? are linearly dependent. Equations (78){(83) may be used to examine the implications of speci c textures of the neutrino mass matrix. In the following, we discuss several speci c textures of the neutrino mass matrix which have been considered in the literature. Their CP e ects are particularly noted.

0 1 2 3 2 2 23 2 23 23 23 12 23 13 23 13 23 12 23 02 23 03 23 03 23 02 23 2 3 0 1 23 0 03 02 23 1 13 12 23 0 1 0 23 0 0 1 0 1 0

A. Me = 0

Speci c examples of this class of models are given in Refs. 14], 19], and 20], in which x = s , y = , and the mass matrices are taken to be real. In these models the mass matrices were chosen to minimize the number of parameters needed to provide the appropriate phenomenology, and a nonzero Me is not required. In Ref. 14] the case x = and y = s was also considered. 13

Using Eq. (80), Me = 0 implies s c ' ?s s , which in turn leads to Ae ; (86) LSND ' 16js j js j sin ey (87) ALSND ' 4js j (1 ? sin 2 sin )=c ; e = ?Aey ' 4js j js j cos 2 ; Aatm (88) atm e = ?B ey ' ?2js j js j sin 2 ; Batm (89) atm and = ?2 . Observable oscillations at LSND requires 6= 0; and 6= 0; . Also, e ! y oscillations in short-baseline experiments, e ! and e ! y oscillations in longbaseline experiments, and CP violation in long-baseline experiments are possible, although not required, in this scenario. Finally, we have q (90) jM j ' jMyy j ' m 1 ? sin 2 sin ; jM y j ' m sin 2 sin : (91) As one example, if we take ! and ! , we obtain the model of Ref. 14], in which there is maximal - y mixing and jM j ' jMyy j jM y j. Furthermore Aey = LSND e ey Batm = ?Batm = 0, so e oscillates only to in short-baseline experiments and there is no visible CP violation in long-baseline experiments. If we allow 6= , we have a model equivalent to that of Ref. 19]; in this case Aey 6= 0 LSND e ey and Batm = ?Batm = 0, so there can be e ! y oscillations in short-baseline experiments but still no visible CP violation in long-baseline experiments. In order to have CP violation in the present case, we must have 6= . Then there must be short-baseline e ! y oscillations, although the existence of long-baseline e $ oscillations depends on the value of . Finally, an interesting case to consider is maximal CP violation (maximal in the sense that it gives the largest CP -violation parameter for a given js j and js j), which corresponds to = . If there is also maximal - y mixing ( = ), then the mass matrix in the - y sector is approximately (after appropriate changes of neutrino phase to make the diagonal elements real to leading order in ) ! ! ! m M M y ' p 1 i + matm ei = 1 1 : (92) 11 My Myy 4m 2 i1 The measurables in short- and long-baseline experiments are then ey Ae (93) LSND = ALSND ' 4js j ; e = ?Aey ' 0 ; Aatm (94) atm e = ?B ey ' ?js j ; Batm (95) atm i.e., e oscillates equally into and y in short-baseline experiments and there are no additional contributions to the CP -conserving part of these oscillations in long-baseline experiments. The vacuum CP and T asymmetries are especially simple in this case, 1 (96) ACP = AT = ?AT ' ? 2 sin 2 atm ; e ey e as the dependence on js j cancels in the ratio. The particular models discussed above are summarized in Table II.

12 23 13 13 23 2 2 23 2 23 13 2 2 23 2 23 2 23 13 2 23 2 23 13 2 23 2 23 0 23 23 23 2 23 2 23 23 23 23 2 23 4 23 4 23 2 23 13 23 23 4 23 4 2 4 13 2 2 13 13 2

14

B. Mey = 0

An example of this class of models with x = s and y = is given in Ref. 36], where a nonzero Mey was not needed to provide the appropriate phenomenology. From Eq. (81), Mey = 0 implies s s ' s c , which leads to

12 23 13 23

Ae LSND ' 4js j (1 ? sin 2 sin ey ALSND ' 16js j c sin ; e = ?Aey ' ?4js j c cos 2 ; Aatm atm e = ?B ey ' 2js j c sin 2 ; Batm atm

13 2 2 23 2 2 13 23 2 2 13 23 2 2 13 23 2 23 23 23 0 23

2

23

)=js j ;

23 2

(97) (98) (99) (100)

23 2 23 4

and = ?2 . The existence of oscillations in LSND implies that 6= or 6= . As with the Me = 0 case, it is possible to have e ! y oscillations in short-baseline experiments, e ! and e ! y oscillations in long-baseline experiments, and CP violation in long-baseline experiments. The approximate magnitudes of the mass matrix elements M , Myy , and M y are the same as given in Eqs. (90) and (91). The limit ! 0 and ' reproduces the model in Ref. 36], which has no e ! y in short-baseline experiments and no visible CP violation in long-baseline experiments. CP violation can occur if 6= 0; ; , in which case there are e ! y oscillations in shortbaseline experiments and there may be e ! and e ! y oscillations in long-baseline experiments, depending on the value of . The Mey = 0 model with maximal CP violation and maximal - y mixing ( = = ) has the same features as the Me = 0 maximal CP violation case in Sec. IV.A, except that the vacuum CP and T asymmetries in Eq. (96) have the opposite sign. The particular Mey = 0 cases discussed here are also summarized in Table II.

23 23 4 23 2 23 4 23 23

C. Me 6= 0 and Mey 6= 0

In this more general case, barring fortuitous cancellations one would expect from Eqs. (59){(68) that there are e ! and e ! y oscillations in short- and long-baseline experiments, and CP violation in e ! and e ! y oscillations in long-baseline experiments. For this texture, jM j; jMyy j jM y j does not necessarily exclude visible CP violation, unlike the cases Me = 0 or Mey = 0. As an example, the mass matrix 0 ei 2 0 0 1 B i2 0 ei 3 C ; C (101) M = mB e B 0 A @ ei 1 C 0 ei 3 ei 1

1 2 2 5 3 5 4 3 6

which is an extension of the model introduced in Ref. 14], leads to CP violation of order in e ! and e ! y oscillations in long-baseline experiments. This mass matrix di ers from the one in Ref. 14] in that the Me and M e elements, denoted as , are not zero, the M and Myy elements, denoted as and , respectively, are not necessarily equal, and the CP -violating phases are not set to zero. The diagonal elements of the mass matrix can be

2 5 4 6

15

taken to be real. Because Mee = Mx = Mxy = 0 there are only three independent phases. The mass eigenvalues and approximate mixing matrix for the mass matrix in Eq. (101) are given in Appendix C. The largest o -diagonal short-baseline oscillation amplitudes in this case are

Ae LSND ' 4 ; ey ALSND ' 4 :

2 3 2 5 4

(102) (103)

Short-baseline - y oscillations are of order . The largest long-baseline oscillation probabilities are

y Aatm ' 1 ; Ae = ?Aey ' ? : atm atm

2 5 2 3

(104) (105)

x

Short- and long-baseline oscillation amplitudes involving observable CP -violating amplitude is

e Batm '

3 5

are of order

4

or smaller. The (106)

sin( + ) ;

3 23 1 3 5 3 5

where (see Appendix C) tan Finally,

23

= ? ( ? ) sin + 2 ( + ) cos + 2

4 6 4 6 1

sin cos

3 3

:

(107) (108) (109)

Aee = sin sun

where tan

01

2

01

;

1 3 2

= +2

2 1

3 5

v u 4 u t1 +

1 3 5

(1 ? cos( + ? 2 )) : ( ?2 )

1 3 5 2

The phenomenology of these models is summarized in Table II.

D. Mee = Me = M = 0

In the special class of models with two sterile and two active neutrinos, i.e., both x and y are sterile, there need not be Majorana mass terms for the two active neutrinos in order to obtain the proper phenomenology. As described in more detail in Appendix B, Mee = Me = M = 0 requires only a minimal extension in the Higgs sector of the Standard Model, i.e., only SU (2) singlet Higgs bosons need to be added. Examples of models with both x and y sterile are given in Ref. 37]. We will now show that CP violation e ects in long-baseline experiments are suppressed in this class of models. Models with Me = 0 have already been discussed in Sec. IV.A; here we add the additional constraint M = 0. It was previously determined that jM j; jMyy j jM y j implied ' and ' , so Eqs. (86){(89) reduce to

23 2 23 4

16

Ae LSND ' 8js j ; Aey ' 0 ; LSND e = ?Aey ' ?2js j ; Aatm atm e = ?B ey ' 0 : Batm atm

13 2 13 2 4

(110) (111) (112) (113)

Hence, there is no visible CP violation in long-baseline experiments in this case (strictly speaking CP violation is strongly suppressed, to order ). Therefore in models with Mee = Me = M = 0, the only phenomenological deviations from the Standard Model are CP conserving neutrino oscillations (see Table II) and the presence of more than one neutral Higgs scalar.

E. Other mass hierarchies

There are other mass hierarchies possible which give the same q oscillation phenomena as those discussed in Secs. IV.A{IV.D. One is m < m m ; m ' mLSND m, in which the solar oscillation occurs between the two upper mass eigenstates and the atmospheric oscillation between the two lower mass eigenstates. Assuming as before that e mixes primarily with x and with y (i.e., s ; s ; s ; s ), then

2 3 0 1 2 02 03 12 13

Me ' m(Ue U + Ue U ) ;

0 0 1 1 2 32

(114)

Mee , Mex, Mxx, Mey , Mx , and Mxy are given by similar expressions with appropriate changes of subscripts. However, for M , M y , and Myy none of the four terms in Eq. (9) are dominant. Since long-baseline ! oscillations involve m , and hence the mixing matrix elements U , U , U , and U (see Eqs. (42){(48)), then a mass texture condition such as Me = 0, when applied to Eq. (114), does not tell us anything speci c about longbaseline oscillations. It could, however, a ect short-baseline oscillation amplitudes, which depend on the U and U (see Eqs. (39){(41)). We do not pursue this possibility further here. Another possible hierarchy is to have m ' m < m ' m where none of the masses are much smaller than the others; in this case, all masses would contribute to hot dark matter (an alternate possibility, m ' m < m ' m with none small, gives similar results). A model of this type has been discussed in Ref. 20]. In this case, again assuming s ; s ; s ; s , we nd

2 3 2 3 0 1 0 1 2 3 2 3 0 1 02 03 12 13

Mex ' m (Ue Ux + Ue Ux ) ;

0 0 0 1 1

(115) (116)

with similar expressions for Mee and Mxx, and

M y ' m (U Uy + U Uy ) ;

3 2 2 3 3

with similar expressions for M and Myy . However, for Me , Mey , Mx , and Mxy , none of the four terms in Eq. (9) are dominant, the expressions for the mass matrix elements are more complicated, and the implications of particular textures for long-baseline oscillations are not as easily determined. We also do not pursue this case further here. 17

V. OTHER PROBES OF NEUTRINO MASS

The presence of mass terms for neutrinos and, in particular, Majorana mass terms, opens up a variety of possibilities for phenomena that are not possible in the Standard Model. The neutrino mass, besides giving rise to mixing of neutrinos and the associated CP e ect discussed in this paper, can lead to lepton avor-changing charged currents analogous to those of the quark sector. Majorana mass can also give rise to lepton number violation processes. With these possibilities, widely searched-for phenomena such as ! e + , ! e + e + e, -e conversion, and electric dipole moments for charge leptons, can occur. Unfortunately, all these processes 38] are proportional to (m =MW ) or ((m =MW )ln(m =MW )) . Given that m is of the order of 5 eV or less 39] these are no larger than 10? and 10? . The current upper bounds 40] are about 30 orders of magnitude larger than those theoretical predictions from the neutrino masses. Therefore they are unobservable. Since this conclusion depends only on the smallness of the neutrino masses, it is valid in general. The Majorana mass term breaks lepton number conservation and can lead to neutrinoless double beta decay. The rate is governed by the magnitude of the e ective e mass X (117) hm e i = j Uej mj j = jMee j ;

4 2 2 4 40 33 2

j

i.e., the magnitude of the Mee element in the Majorana neutrino mass matrix in Eq. (4). The current limit on jMee j from neutrinoless double beta decays is about 0.5 eV 41]. Since

Mee = s m + c m + s m + s m ;

2 01 0 2 01 1 2 12 2 2 13 3 0 1 2 2 12 13 2 3 0 1

(118)

3 2

there q will be no visible neutrinoless double beta decay in models with m ; m m < m ' q mLSND 1 eV and js j js j . In models where m < m m ; m ' mLSND, or if no neutrino masses are 1 eV, neutrinoless double beta decay may provide a strong constraint. Neutrino masses may also a ect cosmology if P m > 0:5 eV 42]. This level of neutrino mass can easily be accommodated by a four-neutrino model with two pairs of nearly degenerate mass eigenstates separated by approximately 1 eV. Because of the smallness of the neutrino masses, there are no other observable e ects besides neutrino oscillations and possibly neutrinoless double beta decay and dark matter. However, the rare decays may still be observable if new physics occurs also in other sectors, such as anomalous gauge boson couplings or anomalous fermion-gauge boson interactions. Therefore, it is important to continue to search for them.

VI. SUMMARY

In this paper we have presented a general parametrization of the four-neutrino mixing matrix and discussed the oscillation phenomenology for the case of two nearly degenerate pairs of mass eigenstates separated from each other by approximately 1 eV, which is the mass spectrum indicated by current solar, atmospheric, reactor, and accelerator neutrino experiments. We analyzed in detail the case where e mixes primarily with x and with y , where one of x and y is and the other is sterile, or both are sterile. We found in these 18

cases that the neutrino mixing matrix can be written in 2 2 block form with small o diagonal blocks. By construction the mixing matrices have e ! oscillations in LSND and ! y oscillations in atmospheric experiments. We found that the following oscillations are also possible: e ! y and ! x in short-baseline experiments, and e ! , e ! y , and ! x (including CP -violation e ects) in long-baseline experiments. We also found that solar, atmospheric, short- and long-baseline oscillation measurements can, in some cases, determine all but one of the four-neutrino mixing parameters. Finally, we examined the implications of some several speci c mass textures, and found the conditions under which CP -violation e ects are visible. As pointed out in the Introduction, additional evidence is needed in order for the fourneutrino scenario to be on rm ground. Given that there must be separate mass-squared di erence scales for the solar and atmospheric oscillations (as currently indicated by the data), there are in fact two ways to verify the existence of four light neutrinos: (i) con rmation of the LSND results, which could occur in the future mini-BOONE collaboration 43{45], or (ii) detection of vacuum (i.e., not matter-induced) CP or T violation in longbaseline experiments, which should be greatly suppressed in a three-neutrino scenario. Once the existence of four neutrinos is established, the next task is to determine the neutrino mixing matrix parameters. We emphasize the signi cant potential for detecting new oscillation channels and CP violation in future high statistics short- and long-baseline oscillation experiments. Many experiments have been proposed and some will be online in the next few years 45,46]. In these experiments neutrino beams are produced at high energy accelerators and oscillations can be detected at distant underground detectors. They include the KEK-Kamiokande K2K Collaboration 47], the Fermilab-Soudan MINOS 48] and Emulsion Sandwich 49] collaborations, and CERN-Gran Sasso ICARUS, Super-ICARUS, AQUA-RICH, NICE, NOE and OPERA collaborations 50]. Experiments done at muon storage rings 46] may be especially important since they will have the ability to measure both e ! and/or , and ! e and/or , as well as the corresponding oscillation channels for antineutrinos. Furthermore, there may also be hitherto undiscovered oscillation e ects in short-baseline oscillation experiments such as COSMOS 51] and TOSCA 52], which will search for ! oscillations. To completely determine all accessible parameters in the four-neutrino mixing matrix requires searches at both short and long baselines. The amplitudes of various oscillation channels, including possible CP violation e ects, will help further determine the texture of the four-neutrino mass matrix and o er a better understanding of neutrino physics as well as CP violation.

ACKNOWLEDGEMENTS

YD would like to thank the hospitality of Iowa State University where this work began. This work was supported in part by the U.S. Department of Energy, Division of High Energy Physics, under Grants No. DE-FG02-94ER40817 and No. DE-FG02-95ER40896, and in part by the University of Wisconsin Research Committee with funds granted by the Wisconsin Alumni Research Foundation. YD's work is partially support by the National Natural Science Foundation of China. BLY acknowledges the support by a NATO collaborative grant. 19

APPENDIX A: GENERAL PROPERTIES OF N N MAJORANA MASS AND MIXING MATRICES

Consider the general case of n neutrinos, in which nR are right-handed (sterile) and nL are left-handed (active), nR + nL = n. We can represent the nR right-handed neutrinos by their left-handed conjugates and denote the collection of all the n independent left-handed neutrinos by a column vector L. Then either Dirac or Majorana neutrino mass terms can be written as (A1) R M L + h:c: ; T with R related to L by R = L C , where C = i is the charge conjugation matrix. The most general n n mass matrix M is symmetric and complex, M y = M , characterized by n(n + 1) magnitudes and the same number of phases. Since a given component eld in L and R acquires the same phase factor under a change of phase, n of the phases in M may be absorbed into the de nitions of the elds, leaving n(n + 1) magnitudes and n(n ? 1) phases; we will often choose the convention that the diagonal elements of M are real and the o -diagonal elements complex. As shown in Eq. (2), the mass matrix may be diagonalized by a unitary matrix U . The e ects of U may be divided into two classes of phenomenology which give rise to violation of individual lepton number: neutrino oscillations and lepton charged currents. Since Z interacts only with the left-handed neutrinos, neutrino counting in Z ! 0 at LEP is unchanged if all of the neutrinos are light, i.e., (N )LEP = nL, with nL = 3 in the Standard Model. This can be demonstrated straightforwardly as follows. The neutral current Lagrangian can be written in terms of avor eigenstates as g L K LZ , where K is an n n diagonal matrix with the rst nR elements being zero and the other nL elements unity. If all neutrinos are very light, which is the case we are considering here, the number of neutrinos measured at LEP is simply Tr(KK y) = nL , assuming the couplings of the active neutrinos are universal. In terms of the mass eigenstates, the neutrino counting is unchanged: Tr(UK (UK )y) = Tr(KK y) = nL. In general an n n unitary matrix such as U can be described by n(n ? 1) rotation angles and n(n + 1) phases. In the charged lepton current, we are free to make phase transformations of the charged lepton elds, which removes n of the phases. Then the number of surviving measurable phases is n(n + 1) ? n = n(n ? 1). This argument is not a ected by the fact that the number of left-handed charged lepton elds, nc , in the charged-current is less than n, as each of the rst n ? nc rows of U can be multiplied by a phase without altering the charged currents. Therefore, in general in this Majorana setting we can parametrize U by n(n ? 1) angles and n(n ? 1) phases. In neutrino oscillations, however, only (n ? 1)(n ? 2) independent phases can in principle be measured, as we will now demonstrate. Note that the W jk in Eq. (11) are invariant when U is transformed from either the left or right side by a diagonal matrix which contains only phases, i.e., U j ! ei U j ei j : (A2) Then, as far as neutrino oscillations are concerned, we can eliminate 2n ? 1 of the phases in U , so that there are e ectively only

2 0 1 2 1 2 1 2 0 0 1 2 1 2 1 2 1 2 1 2 1 2 1 2

20

1 n(n + 1) ? (2n ? 1) = 1 (n ? 1)(n ? 2) ; (A3) 2 2 independent phases that can be measured by neutrino oscillation experiments. Interestingly this is the same number of independent phases that may be determined in the CKM matrix for n generations of quarks. However, note that in the most general U there are n(n ? 1) phases, so that there are n(n ? 1) ? (n ? 1)(n ? 2) = (n ? 1) phases that cannot be determined from neutrino oscillations. The phase counting of Eq. (A3) can also be con rmed by enumerating the number of independent CP -violating variables, e.g., the P de ned in Eq. (22), that can be measured. There are n(n ? 1) such di erences, but from Eq. (15) P = ? P and from Eq. (19) P P = 0, so it follows that n ? 1 of the di erences are not independent. Therefore there are only (n ? 1)(n ? 2) independent P , and only (n ? 1)(n ? 2) CP -violation parameters can be measured.

1 2 1 2 1 2 1 2 1 2 1 2

APPENDIX B: HIGGS BOSON ORIGINS OF NEUTRINO MASSES

The presence of masses for neutrinos is a de nite signal of physics beyond the Standard Model. Particularly, with the three types of neutrino oscillations which indicate three m scales and require at least four neutrino mass values, a non-trivial extension of the Standard Model is necessary. In searching for hints of the extension, it is interesting to consider what simplest extensions of the Standard Model are possible and how natural (or unnatural) they are in their couplings schemes. In this appendix we discuss brie y the possible origins of the two types of mass matrices considered, i.e., models with one or two sterile neutrinos. Then both Dirac (active-sterile) and Majorana (active-active or sterile-sterile) neutrino mass terms are present. These masses can be obtained by suitable extensions of the Standard Model. We will only enlarge the lepton Yukawa sector and the Higgs sector to the extent required by the mass matrices of Sec. IV. Our goal is to assure that such mass matrices are possible by straightforward modi cations of these two sectors, and to determine what new particles need to be added to the Standard Model spectrum. We do not attempt to construct the best case scenario, which can be done when more information on the neutrinos mass are available. For a more extensive discussion of possible origins of neutrino masses terms, see Ref. 53]. In the following we denote the right-handed sterile by sjR and the corresponding leftT handed conjugate by ^sjL = sjR C . We also denote the left-handed lepton SU(2) doublet T by lkL with the corresponding right-handed conjugate elds ^kR = ?i C lkR, j and k are l generation labels.

2 2

1. Models with two left-handed and two right-handed neutrinos

We rst consider the class of models in which x = s and y = s , where s and s are the right-handed sterile neutrinos that are associated with e and , respectively, which have been considered in Ref. 37]. We discuss two cases: (i) models where only the sterile neutrinos have nonzero Majorana mass terms, and (ii) models where both right-handed (sterile) and left-handed ( e and ) elds have Majorana mass terms. In the rst case

1 2 1 2

21

Mee = Me = M = 0, which is discussed in Sec. IV.D; in the second case, Me 6= 0, which can be realized in the models discussed in Secs. IV.B and IV.C. The simplest extension of the Standard Model is case (i) above, which can be obtained by adding a singlet real scalar eld to the Standard Model Higgs doublet . The Majorana masses for the two sterile are due to their coupling to and are proportional to the vacuum expectation value (vev) of . The provides the Dirac masses from Yukawa couplings involving both sterile and left-handed neutrinos. We denote the absolute value of the vev's of the and elds as v and V respectively; v is the same as the Standard Model vev. The Yukawa couplings can be written as X X LY = Gjj sjR ^sj L + gjk ( sjR ~ ylkL + l^kR ^sR) + h:c: ; (B1)

2

j;j

0

0

0

j;k

where Gjj and gjk are complex couplings, and ~ = i . Since sjR ^sj L = ^sj L sjR we have Gjj = Gj j . We also exhibit the symmetry of the Dirac coupling coe cients, gjk = gkj , because of the identity sjR ~ ylkL = ^kR ^sjR l For the Higgs potential, we take the simpli ed case that it has a Z symmetry in , i.e., invariant under ! ? . Then the Higgs potential contains only real coe cients:

0 0 0 0 0

2

LH = ? j j ?

2 1 2

2 2

2

+ j j +

4

2

4

+

3

2

j j:

2

(B2)

After spontaneous symmetry breaking there are two massive neutral Higgs bosons. Their masses are set by v and V . The value of v is the same as in the Standard Model. The coe cients of the Yukawa couplings must be very small in order to give neutrino masses of the order of eV. In the case V v, one of the Higgs boson is composed mostly of the neutral eld of with a mass proportional to v and the other composed of mostly the eld with a mass proportional to V . There are no other changes to the Standard Model phenomenology. One can also have a more complicated scenario of case (ii) in which the Majorana masses of the two left-handed neutrinos are non-vanishing. These types of models can be constructed by the approach discussed below in Appendix B.2.

2. Models with three left-handed and one right-handed neutrinos

Here we consider the cases (i) x = s and y = , or (ii) x = and y = s , where s is a sterile neutrino. In the rst case, mass terms are needed in the - sector to provide the large mixing of atmospheric neutrinos, while in the second case Majorana mass terms are needed to provide mixing of solar neutrinos. There are no constraints on which terms in the mass matrix may be nonzero, but in each case Majorana masses of the left-handed (active) neutrinos must exist. The phenomenology of some of these models is discussed in Secs. IV.A{IV.C. Majorana mass terms between left-handed neutrinos can arise from the introduction of a Higgs triplet which has lepton number ?2 54]. The Majorana mass of the sterile neutrino again comes from a Higgs singlet as discussed above. We denote the triplet by 22

1 = p2 ~ ~ =

0 2

+

=p2

0

? = p2 ;

++ +

!

(B3)

0

and the value of the vev of by . Since contributes to the masses of the W and Z bosons di erently, it has to be small in comparison with the vacuum expectation of the , say =v < 10? , so that the bulk of the electroweak gauge boson masses come from the doublet. Then, this will not upset the good agreement achieved by the Standard Model prediction for the parameter. The Yukawa couplings can be written as X X l (B4) LY = Gs sR ^sL + gk ( sR ~ ylkL + ^kL sR) + hkk ^kR lk L + h:c: ; l

k k;k

0 0 0

where Gs , gk and hkk , k and k0 = 1; 2; 3, are complex couplings. The symmetry of the Dirac couplings is explicitly exhibited because of the identity sR ~ ylkL = ^kL sR. hkk is l ^kR lk L = ^k R lkL. For the Higgs potential, l symmetric, hkk = hk k , because of the identity l we can again take the simpli ed case that is a real scalar eld and the Higgs potential has the Z symmetry in : LH = ? j j ? ? Tr( y) + y ~ + ~ y y + j j + + (Tr( y)) + Tr(( y) ) + Tr( ( y) ) + Tr Tr(( y) ) + j j + Tr( y) + y y + y y + j j Tr( y) : (B5)

0 0 0 0 0 0

2

2 1

2

2 2

2

2 3

1

4

2

4

3

2

4

2

5

2

2

6

2

2

1

2

2

2

2

3

4

5

2

The following terms in the Higgs potential, y ~ + ~y

y

;

(B6)

are needed to break the global lepton number invariance in order to avoid the appearance of a Goldstone boson called Majoron 54], and is the only coupling that potentially can be complex. To obtain the neutrino mass matrix we can also make phase tranformations on the fermion elds sR and lkL to make Gs and the diagonal terms hkk real. If CP is not broken spontaneously, which we assume to be the case here, the vacuum expectation values of all neutral elds can be made real by phase transformations on the Higgs elds and . After spontaneous symmetry breaking this choice of phases for the Yukawa couplings agrees with the convention of the neutrino mass matrix discussed in Sec. II. With complex couplings, CP violation can generally occur in the Higgs sector. However, is required to be real by the minimization of the Higgs potential. Hence explicit CP violation does not occur in this extended Higgs scenario. A more complicated Higgs potential can be chosen to allow complex couplings so that CP violation can be manifest in the Higgs sector. We will not elaborate on this possibility here. After spontaneous symmetry breaking, the physical Higgs boson spectrum contains a doubly charged pair, a singly charged pair, and four neutrals. The Goldstone bosons are mostly from the Higgs doublet . The masses of two of the neutral Higgs bosons are proportional to v. The masses of the remaining two neutral Higgs boson are similar to those of the case of Appendix B.1. Again the Majorana couplings of the sterile neutrino and the 23

Dirac coupling of the sterile to the left-handed neutrinos are small. However, the Majorana couplings among the left-handed leptons do not have to be small if is chosen to be the order of the neutrino masses, i.e., eV 55]; this can be done without leading to any small Higgs boson masses. Then the coupling of the singly and doubly charged Higgs boson to the charged leptons are not small. The production of these particles in a future high energy linear collider or muon collider is possible if they are not too heavy. In this type of model, since the constraints Mee = Me = M = 0 do not apply, the CP violating parameter Im(Ue Ue U U ) is no longer constrained to be approximately zero. The CP violation can be of order , which is the same order as the oscillation probabilities themselves, and hence measurable. Examples of this type of model include the maximal CP violation models characterized by Eq. (92) and the mass matrix in Eq. (101). We note that to produce the required neutrino masses and the phenomenologically interesting mass spectrum and expectation values of the Higgs boson elds in both models dicussed in this section, new hierarchy problems are introduced 55]. In our view, such hierarchy problems do not necessarily argue against the models. However, it does argue that any model of this sort should be included in a larger, more natural, scheme. Note that although the hierarchy in the expectation values of the Higgs boson elds sometimes leads to a ne tuning of the parameters, the small vacuum expection value of may be obtained in a natural way 56]. For example, if v ;V ; , then the minimization of the Higgs potential leads to the relation ' ? v = v. There is a growing literature on the generation of neutrino masses. An intriguing class of models are those that generate mass dynamically by higher order loop e ects 57]. We refer the reader to Ref. 58] for recent and extensive analyses of this possibility for four neutrinos. There are also models that use lepton-number violating interactions in R-parity violating supersymmetry for the generation of Majorana mass 59].

2 3 2 2 3 3 2 2 2 2 2 3

APPENDIX C: AN EXAMPLE WITH CP VIOLATION

In this appendix we derive the masses and mixing matrix for the model described by the mass matrix in Eq. (101). In general a 4 4 Majorana mass matrix can have six independent phases (see Appendix A), but since three of the mass matrix elements are zero, there are only three independent phases in this case. We have chosen to make the diagonal elements of M real. To achieve the proper neutrino phenomenology, we assume the following hierarchy ; ; ; 1: (C1) The mass-squared eigenvalues are approximately given by m ' m ; m ' 4 m ; m ; ' (1 + + )m ; (C2) where =4 +( ? ) +4 c +4 cos( ? ) + cos( + )] ; (C3) with cj cos j and sj sin j . The eigenvalues are related to the physical mass-squared di erences by

2 1 4 6 3 5 2 0 2 1 2 2 1 2 2 3 5 2 2 23 2 3 2 5 2 0 2 4 0 2 2 3 5 4 6 2 2 4 6 1 3 5 4 1 3 6 1 3

24

mLSND = m ? m ' m ; matm = m ? m ' 2 m ; msun = m ? m ' ( ? 4

2 2 2 2 2 2 3 2 0 2 1 2 2 2 1 2 2 0 2 1 2 2 2 0 2 1

2 2 3 5

)m :

2

(C4) (C5) (C6)

The size of msun in Eq. (C6) implied by the hierarchy of Eq. (C1) means that the solar neutrino oscillations are of the MSW type. Therefore in order to have the proper MSW enhancement in the sun we must have m > m , which implies j j > j2 j. The matrix U that diagonalizes M via Eq. (2) is given approximately by 0 1 c s e?i 01 0 0 i i B ?s ei 01 ep 1 ( e?i 3 ? ei 23 ) ep 1 ( e?i 3 23 + ) C c B C B C; U ' B s ei 01 3 ? 1 ? c ei 3? 1 C p p e?i 23 @ A i 01? 1 ?i 1 p ei 23 p s e ? c e ? (C7)

1 3 5 01 01 01 01 3 01 ( + ) 3 01 ( ) 2 3 5 5 01 ( ) 5 01 1 2 1 2 2 3 ( + 5 1 2 1 2

where

with c

cos , s

( ? ) sin = ? ( + ) cos + 2 s ; +2 c 2 s = tan? ?2 c ? ; sin , ? ? 2 , and v u 4 u tan = + 2 t1 + ( ? (1 ? c ) : 2 ) tan

23 4 6 1 3 5 3 4 6 1 3 5 3 01 1 3 5 1 3 5 2 3 1 2 01 2 1 3 5 1 1 3 5 3 5 2

(C8) (C9)

(C10)

We note that this U has the form of of Eq. (56). It also can be seen to have the form of Eq. (58) if we set

02

=

03

= 0;

23

and make the identi cations

13

= 4;

(C11) (C12) (C13)

1 s e?i 13 = p ( e?i 3 ? ei 23 )ei 1 ; 2 1 ( e?i 3 23 + )ei 1 : s e?i 12 = p 2

3 5 12 3 ( + ) 5

25

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29

TABLE I. Parameters in the four-neutrino mixing matrix and the primary observables used to determine them. Parameter(s)

01

, 13, 02 , 03 ,

12

23 1 0 13 03

? ?

12 02

? ?

23 23

Aee sun y Aatm e Ae ,Ae ,Batm LSND atm x ,A x ,B x ALSND atm atm

Primary Observable(s)

s02; s03; s12; s13

TABLE II. Summary of some particular four-neutrino models for m0 ; m1 m2 < m3 and . All models in the table have been constructed to have short-baseline ! e ocsillations in agreement with the LSND data and large-amplitude ! y oscillations in atmospheric and long-baseline experiments; they also all have negligible e ! x and ! y oscillations in short-baseline experiments. The size of ! x oscillations and CP violation in long-baseline ! y oscillations depend on other model parameters. In all cases, one of x and y could be and the other sterile, or both could be sterile. CP-conserving CP-conserving CP-violating short-baseline long-baseline long-baseline

23 2 23

Texture Me = 0

e! y

6=

Mey = 0

2 2

6= 4 any

4

4

6= 0; 2 ;

4

0

4

6=

4 2

Me ; Mey 6= 0 varies Mee = Me = M = 0 2

4 4 4

No Yes Yes Yes Yes Yes Yes Yes No

Yes Yes Maybe No No Maybe No Yes Yes

e

!

e!

No No Yes Maximal No Yes Maximal Yes No

;

y

Reference Ref. 14] Ref. 19] Sec. IV.A Eq. (92) Ref. 36] Sec. IV.B Eq. (92) Eq. (101) Sec. IV.D

30

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