Scholarly article on topic 'Minimal seesaw model with tri/bi-maximal mixing and leptogenesis'

Minimal seesaw model with tri/bi-maximal mixing and leptogenesis Academic research paper on "Physical sciences"

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Abstract of research paper on Physical sciences, author of scientific article — Sanghyeon Chang, Sin Kyu Kang, Kim Siyeon

Abstract We examine minimal seesaw mechanism in which the masses of light neutrinos are described with tri/bi-maximal mixing in the basis where the charged-lepton Yukawa matrix and heavy Majorana neutrino mass matrix are diagonal. We search for all possible Dirac mass textures which contain at least one zero entry in 3 × 2 matrix and evaluate the corresponding lepton asymmetries. We present the baryon asymmetry in terms of a single low energy unknown, a Majorana CP phase to be clued from neutrinoless double beta decay.

Academic research paper on topic "Minimal seesaw model with tri/bi-maximal mixing and leptogenesis"

Available online at www.sciencedirect.com

SCIENCE ^DIRECT'

Physics Letters B 597 (2004) 78-

www.elsevier.com/locate/physletb

Minimal seesaw model with tri/bi-maximal mixing and leptogenesis

Sanghyeon Changa, Sin Kyu Kangb, Kim Siyeonc

a Department of Physics, Yonsei University, Seoul 120-749, South Korea b School of Physics, Seoul National University, Seoul 151-734, South Korea c Department of Physics, Chung-Ang University, Seoul 156-756, South Korea

Received 27 April 2004; received in revised form 24 June 2004; accepted 29 June 2004

Available online 14 July 2004

Editor: T. Yanagida

Abstract

We examine minimal seesaw mechanism in which the masses of light neutrinos are described with tri/bi-maximal mixing in the basis where the charged-lepton Yukawa matrix and heavy Majorana neutrino mass matrix are diagonal. We search for all possible Dirac mass textures which contain at least one zero entry in 3 x 2 matrix and evaluate the corresponding lepton asymmetries. We present the baryon asymmetry in terms of a single low energy unknown, a Majorana CP phase to be clued from neutrinoless double beta decay. © 2004 Elsevier B.V. All rights reserved.

PACS: 14.60.Pq; 14.60.St; 13.40.Em

1. Introduction

Thanks to the accumulating data from experiments on the atmospheric and solar neutrinos experiments [1-3], we are now convinced that neutrinos oscillate among three active neutrinos. Interpreting each experiment in terms of two-flavor mixing, the mixing angle for the oscillation of atmospheric neutrinos is understood to be maximal or nearly maximal: sin2 20atm ~ 1, whereas the one for the oscillation of solar neutrinos is not maximal but large: sin2 0sol ~ 0.3 [6]. The upper bound for 0reac, sin0reac < 0.2, was obtained from the non-observation of the disappearance of ve in the CHOOZ experiment [4] with Am2 < 10-3 eV2. The unitary mixing matrix is defined via va = J23=1 UajVj (a = e, x, t), where va is a flavor eigenstate and vj is a mass eigenstate. Including data from

E-mail addresses: schang@phya.yonsei.ac.kr (S. Chang), skkang@phya.snu.ac.kr (S.K. Kang), siyeon@cau.ac.kr (K. Siyeon).

0370-2693/$ - see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.physletb.2004.06.104

SNO [3] and KamLAND [5], the range of the magnitude of the MNS mixing matrix is given by [7-10],

/0.79-0.86 0.50-0.61 0-0.16 \ \U | =1 0.24-0.52 0.44-0.69 0.63-0.79 I (1)

V0.26-0.52 0.47-0.71 0.60-0.77/

at the 90% confidence level. The existing data also show that the neutrino mass squared differences which induce the solar and atmospheric neutrino oscillations are Am2ol — (7+0) x 10-5 eV2 and Am^lm — (2.5—¿"9) x 10-3 eV2, respectively. It can be readily recognized that the central values of elements in the mixing matrix in Eq. (1) are pointing an elegant form, which is called tri/bi-maximal mixing [11],

/ A 0 ^

v/6 v/3 v/2

V_J_ J_ J_ ,

\ .if .ñ. ./? /

v/6 v/3 v/2

which consists of sin0soi = l/\/3 and sin0atm = l/x/2. [12] of

the mass matrix based on the particular mixing type ^TB .

On the other hand, the baryon density of our Universe i2Bh2 = 0.0224 ± 0.0009 implied by WMAP (Wilkinson Microwave Anisotropy Probe) data indicates the observed baryon asymmetry in the Universe [13,14],

);CMB = r^ns = x 10_I0i (3)

where nB,nB and nY are number density of baryon, anti-baryon and photon, respectively. The leptogenesis [15] has become a compelling theory to explain the observed baryon asymmetry in the Universe, due to increasing reliance on the seesaw mechanism from experiments. Theory for lepton asymmetry requires two heavy right-handed neutrinos or more. For that reason, a class of models with two heavy right-handed neutrinos and 3 x 2 neutrino Dirac mass matrix is called the minimal neutrino seesaw models (MNSMs) which were intensively studied by several authors recently [16,17], especially for simple Dirac mass textures that make prediction compatible with solar and atmospheric neutrino data.

The main framework of our work is seesaw mechanism in bottom-up approach. We launch our analysis by taking ^TB for mixing of light neutrinos and then investigate the structure of 3 x 2 Dirac matrix. That is, our concern remains on the combination of tri/bi-maximal mixing and MNSMs, in order to study the phenomenological implication of the high energy theory based on the low energy theory. One advantage of our framework is that physical observables can be explained in minimal terms of physical parameters. In Section 2, we present the light neutrino mass matrix in terms of the mixing given in Eq. (2) and mass squared differences measured in experiment. The mass matrix reconstructed in that way will constrain Dirac mass matrix. In subsections, depending on the type of mass hierarchy, possible 3 x 2 Dirac matrices will be examined carefully. In Section 3, leptogenesis will be discussed in details based on the Dirac matrices investigated before. In Section 4, we will present numerical results on leptogenesis in our scheme and a relationship between leptogenesis and neutrinoless double beta decay as well as the lower bound of M1 will be discussed focusing on the contribution from a single Majorana phase.

2. Dirac mass matrices in minimal seesaw

In general, a unitary mixing matrix for 3 generations of neutrinos is given by

Ü = R(023)R(0i3, 8)R(di2)P(<p, <'), (4)

where R's are rotations with three angles and a Dirac phase 8 and the P = Diag(l,e!</2,e!</2) with Majorana phases < and < is a diagonal phase transformation. The mass matrix of light neutrinos is given by

Mv = UDiag(m1,m2,m3)UT, where m1, m2, m3 are real positive masses of light neutrinos. Or the Majorana phases can be embedded in the diagonal mass matrix such that

Mv = UDiag(mi, ni2, ni3)UT,

where U = UP—1 and m2 = m2el<p and m3 = m3e'p .

If the UTB in Eq. (2) is adopted for the U in Eq. (5), the light neutrino mass is

/1 0 0 \ /1 1 1 \ / 0 0 0s

/ \ m 2 — mi \ m 3 — mi Mv = m 1 0 1 0 + —-L 1 1 1 + —-L 0 1 -1

,0 0 L

\0 -1 1

which orients toward a minimal model of neutrino sector by removing an angle and the Dirac phase. With SNO and KamLAND, data have narrowed down the possible mass spectrum of neutrinos into two types, Normal Hierarchy (NH), m1 < m2 <m3, and Inverse Hierarchy (IH), m3 < m1 < m2 for MSW LMA. Those two types include the possibility of zero mass for a neutrino, which is necessarily followed by relegating one of the Majorana phases to the unphysical. In other words, the minimal model with the physical observables which the present experimental data guarantee can be obtained by and dictating zero mass to one generation of neutrinos, where the non-zero physical parameters in the model consist of 2 masses, 2 angles, one Majorana phase.

When only two of three neutrinos are massive, by accommodating the experimental results Am2ol = m2 - m^

and Am 2tm =

— m21 to the two types of mass hierarchies, one can obtain the following expressions for mass

eigenvalues:

m 1 = 0, m2 = y A/«2ol, m3 = J A m;ol + A m2atm, m 1 = y A mltm - Am;oV m2 = J A m^, ;«3 = 0,

forNH.

for IH.

Phase transformation P = Diag(1,e!p/2, 1) now can replace the phase transformation in Eq. (4) without loss of generality, whether NH or IH, so that one can single m2 out in order to investigate the CP-violating contribution of the Majorana phase.

Effective neutrino mass models with one zero mass eigenvalue involved in three active neutrinos can be generated naturally from MNSMs. In the basis the mass matrix MR of right-handed neutrinos NR = (N1,N2) is diagonal, the model is given

£ = -JLML1R - vLmDNR + ^ÑcrMrNr + h.c.,

from which the light masses are derived through the seesaw mechanism, Mv = —moM-lmTD in top-down approach. On the other hand, the matrix mD is found as the solution to the seesaw mechanism in bottom-up approach once one launches the analysis with the light neutrino masses Mv. Let M1 and M2 be the masses of right-handed neutrinos and Mn the elements of the matrix Mv. The Dirac matrix,

s[M[a\

mD = I s/M\a2

\s/M[a3 is resulted in with

a\ = y Mu ~bj, bi = y Mu

M11M23 =

aiMii - oibiJ MnMii - Mft

biMii + OiaUMnMii - M2

M12M13 + 0203J(M11M22 - M22) (M11M33 - M

where the i is 2 or 3, the at is a sign ±, and the sign of a1 is fixed as positive. The solution in Eq. (12) was first derived and formulated in Ref. [17]. It is clear that only 5 parameters out of 6; a1, b1, at, bt, can be specified in terms of the elements of Mv. There are various ways to decrease the number of parameters in Dirac matrix, posing one or more zeros or posing equalities between elements for the matrix texture. It is known that texture zeros or equalities among matrix entries can be generated by imposing additional symmetries to the theory.

In this Letter, we focus on posing one or more zeros in Dirac matrix, and show that only one-zero textures are allowed for NH and only one-zero and two-zero textures are allowed for IH, accompanied with the low energy mixing UTB. On the other hand, from Eq. (11), one can recognize that, if there exists a kind of symmetry between entries in Mv, the Dirac matrix also has a symmetry in certain entries inherited from the symmetry of the Mv. So, there are a number of patterns with equalities among the entries in Dirac matrices obtained with one or two zeros, as a consequence of maximal mixing.

2.1. Normal hierarchy

With m1 = 0, the neutrino mass Mv in Eq. (6) reduces to m /1 1 1 \

Mv = — | 1 +d 1-d), (12)

3 V 1 + d)

where d = 3m3/2mi2, which, using Eq. (11), gives rise to Dirac matrix with the following entries:

a\ = tJrii2/3 - b\ = tJrii2/3 - a\,

a\ = a\ — a\b\\fd, bj =b\ + o\ci\\fd, /=2,3, (13)

where a2a3 = -1. Depending on the position of texture zero, the types of Dirac matrices can be classified as follows:

• NH 1-a: &i=0, a\ = y/m 2/3, a\=ci2 = a3, = — = ^yjm^/l',

• NH 1-b: a1 = 0, a1 ^ b1, at ^ bt inNH 1-a;

• NH 2-a: b2 = 0, a2 = V;«2/3 + ;«3/2,

«2 a2(l-d) b\ r

a\ = ---, «3 = —--—, — = -a2s/d, ¿73 = 2&i;

1 + d 1 + d a1

• NH 2-b: a2 = 0, a1 ^ b1, at ^ bt in NH 2-a.

The matrix in Eq. (12) features the equalities between M22 and M33 and between M12 and M13 as a consequence of the maximal mixing of atmospheric neutrinos. In the case with b2 = 0, it can be recognized that the ratios of a1 to a2 and a3 to a2 inherit those of M12 to M22 and M23 to M22, respectively. b3 = 0 or a3 = 0 cases will not be presented as an independent case, since it can be made by exchanging b3 with b2 and a3 with a2 from NH 2-a and 2-b.

2.2. Inverse hierarchy With m3 = 0,

/ x + 2 x — 1 x — 1\

Mv = ^( x+1/2 x+1/2), (14)

x + 1 /2

where x = m 2/mi, which, using Eq. (11), gives rise to Dirac matrix with the following entries:

a\ = yjm\(x + 2)/3 — b2, b\ = ^m\(x + 2)/3 — a2,

= ((,t - l)fli -3abiy/x/2)/(x+ 2), bj = ((„r - l)bi + 3aeny/x/2)/(x + 2), /=2,3, (15)

where a = a; and a2a3 = 1. The equality M22 = M23 = M33 in Eq. (14), which is again a consequence of the maximal mixing of atmospheric neutrinos, constrains the elements of the Dirac matrix such that a2 = a3, b2 = b3. Hence, texture with a single zero included appears only if a1 = 0 or b1 = 0, while texture with two zeros appears

if a2 = a3 = 0 or b2 = b3 = 0.

rz--a\(x — 1) bj 3 a sjx /2

• IH 1-a: b i=0, «i = y ;«2/3 + 2;«i/3, =-, — =-;

x + 2 ai x — 1

• IH 1-b: a1 = 0, a1 ^ b1, a; ^ b; inlH 1-a;

rz--2a2(x — 1) b\ 3a s/x/2

• IH 2-a: b2 = b3= 0, a2 = a3 = y ;«2/3 + ;«i/6, a\ =-, — =--;

2x + 1 a1 x — 1

• IH 2-b: a2 = a3 = 0, a1 ^ b1, a; ^ b; in IH 2-a.

Listed are all the cases with one or more texture zeros in Dirac matrix derivable from the light neutrino mass with UTB, whether NH or IH. In the following, the eligibility of each case to generate the CP asymmetry for leptogenesis will be examined.

3. Leptogenesis

The baryon asymmetry Eq. (3) can be rephrased

HB — nS 1 1

YB = —-~ (8.8-9.8) x 10"11. (16)

The nY is the photon number density and the s is entropy density so that the number density with respect to a co-moving volume element is taken into account. The baryon asymmetry produced through sphaleron process is related to the lepton asymmetry [18,19] by

YB=aYB-L =-tYl, (17)

a — 1

where a = (8NF + 4Nh)/(22Nf + 13Nh), for example, a = 28/79 for the Standard Model (SM) with three generations of fermions and a single Higgs doublet, NF = 3, NH = 1. The purpose of this work is to estimate whether the Yukawa interaction which produces the light neutrinos with the mixing Eq. (2) through the seesaw mechanism can also generate a sufficient lepton asymmetry for the observed baryon asymmetry. The generation of a lepton asymmetry requires the CP-asymmetry and out-of-equilibrium condition. The YL is explicitly parameterized by two factors, e, the size of CP asymmetry, and k , the dilution factor from washout process.

nj — ^ e;

YL = —-(18)

where g* ~ 110 is the number of relativistic degree of freedom. The e; is the magnitude of CP asymmetry in decays of heavy Majorana neutrinos [20,21],

r(Nj IH) - r(N, IH*) £' r(Nj IH) + r(Nj IH*)

where i denotes a generation. When one of two generations of right neutrinos has a mass far below that for the other generation, i.e., M1 < M2, the ei in Eq. (19) is obtained from the decay of M1 [22-24],

&Tf2 (mlmD)n \Mj

where v = 174 GeV and f(M2/M1) represents loop contribution to the decay width from vertex and self energy and is given by

f(x)= x

l-(l+,:2)ln-^ +

l+x2 1 '

1 - „r2

for the Standard Model. For large value of x, the leading order of f(x) is (-3/2)x 1.

It is convenient to consider separately the factor that depends on Dirac matrix in e1 in Eq. (20) at this stage.

Im [{m^mof^ Im[(a*fei + a%b2 + fl3*&3)2] ....

= M2-:—pr——;—n-= M2A\, (22)

(mDmD) 11 l«112 + |a2|2 + l«312

where a's and b's are defined in Eq. (11). From a number of types of matrices with a texture zero derived in Eq. (13) and Eq. (15), only 6 different non-zero values of A1 's can be evaluated. Those particular Dirac matrices to contribute the imaginary parts are the matrix with b2 = 0 and that with a2 = 0 for NH, and the matrix with b1 = 0, that with a1 = 0, that with b2 = b3 = 0, and that with a2 = a3 = 0 for IH. For NH, if a1 = 0, or b1 = 0, the (mDmD)12 vanishes from the trivial relation between entries. Applying Eq. (13) andEq. (15) forEq. (22), one can find that each type of Dirac matrix gives rise to A1 as follows:

Zii(NH 2-a) =---^(^-»»Driny (23)

(2m2 + 3m2)*/4m2 + 9m2 + 12m2m3 cosp —6(m2 — mi) sinp

4i(NH2-b) = — 2 - (24)

+ 9/m2 + Ylm2m3 cos (p where m2 and m3 are given in terms of Am2ol and Am2tm in Eq. (7)

A (IH 1 ) —2;«i;m2(;«t — ) siny ^^

(2;m2 + + m\ + 4;«i;«2cosy

2(ml — m2) sin<p

Z\i (IH 1-b) = — 21 (26)

4;« 2 + m2 + 4m1m2 cos <<

A (IH 2 ) —2;«i;m2(;«t — ) siny ^^

(m 2 + 2m2,)J m2 + 4m2 + 4m1m2cos <p

2(ml — m2) sinp

Zii(IH2-b) = — 21 (28)

3 tJih2 + 4/M2 + 4;«I;M2 cosy

where m1 and m2 are given in terms of Am2ol and Am;;tm in Eq. (8). Thus, for M2 > M1 case, the CP asymmetry in Eq. (20) reduces to e1 « 3/( 16nv2)M1 A1, which is now parameterized by the lightest mass of heavy neutrino M1 and Majorana phase p. The sign of e1 depends on the position of a texture zero in a row of Dirac matrix.

The k in Eq. (18) is determined by solving the full Boltzmann equations. The k can be simply parameterized in terms of K defined as the ratio of A the tree-level decay width of N1 to H the Hubble parameter at temperature M1, where K = r1/H < 1 describes processes out of thermal equilibrium and k < 1 describes washout effect [19,25],

forl0</:< 106, (29)

K(ln K)06

k--1 for 0 < K < 10. (30)

The decay width of N1 by the Yukawa interaction at tree level and Hubble parameter in terms of temperature T and the Planck scale Mpl are A = (m^DmD)11M1/(8nv2) and H = 1.66g1/2T2/Mpi, respectively. At temperature T = M1, the ratio K is

K_ mp1 imDmp) n (31)

1.66^(87102) Ml

which reduces to, using the Dirac matrices in Eq. (10),

10~ eVv

where all fixed numbers are included in a factor of order. As done for the ^1's, one can apply Eq. (13) andEq. (15) for Eq. (32) to find dilution factor k when the decay width is determined by Yukawa couplings in each type of Dirac matrix. For the six types of Dirac matrices that are eligible for the CP asymmetry as in Eqs. (23)-(28), the ratio K for each case is

(2m2 + 3m?, 5mrm3)

i(NH2-a, 2-b)«--" (33)

0—3 eV) 4m2 -L 9m2 .

** _3 (\ai\2+\a2\2+\a3\2), (32)

(10~3 eV)^;«? + 9;«? + 12;m2>M3 costp

(2m2 + m2, 3m1m2)

£(IHl-a, 1-b)«-12 (34)

0—3 eV) 4m2 m2

(10~3 2 + + 4;«i;«2cosy

(m2 + 2m2, 3m1m2)

£(IH 2-a, 2-b)«- ) (35)

(10~3 QV)^m2 + 4;«? + 4;«i;«2cos(p

which shows that the dilution factor also depends on the phase y, but it does not significantly affect the order of magnitude. Out of all the types of Dirac matrices examined, there is no such a case that Yukawa couplings originate decays of neutrinos N1 which satisfy the out-of-equilibrium condition K < 1 at T = M1. The washout effect of asymmetry is most suppressed with the Dirac matrix of type NH 2-b, where, depending on y, the dilution factor ranges from 0.010 to 0.013, the amount of asymmetry survived from washout is at most about 1%. When T < M1, the Boltzmann equations still depict the finite value of k as M1/T increases for the Universe evolution [20,21,26].

4. Discussion

Based on the formulation of the leptogenesis derived in the previous section, we numerically analyze baryon asymmetry for each case classified as NH or IH. For the numerical calculation, we take Am2ol = 7.0 x 10—5 eV2 and Am2tm = 2.5 x 10—3 eV2 as inputs.

Consider a model with neutrino masses in normal hierarchy. In Fig. 1, we plot the baryon asymmetry YB as a function of the Majorana phase y for NH 2-b. The different curves correspond to M1 = 2.0 x 1011 to 2.0 x 1013 GeV for fixed M2/M1 = 5. We note that we can choose any reasonable M2/M1 value which can protect

2.0 x 10^ GeV 2.0 x 10 ; GeV 2.0 x 10 GeV

1e-0 7

1e-10 h

0 20 40 60 80 100 120 140 160 180

Fig. 1. YB as a function of Majorana CP phase for case NH 2-b, with various values of M\ where M2/M\ = 5. The horizontal lines are the current cosmological bound of Yb .

L-violating processes with N1 from the wash-out when T < M2. As expected from Eq. (20), the value of YB for a fixed y increases with M1. The horizontal line in Fig. 1 presents the current cosmological observation of YB given in Eq. (16). From the analysis, we see that the current observation on YB constrains the lower bound of M1, which turns out to be M1 > 2.0 x 1011 GeV. It is clear that the CP asymmetry in high energy is almost proportional to the imaginary part of Majorana CP contribution in low energy from Eqs. (22)-(28). Thus, the plots show that the lower bound of M1 to generate the observed baryon asymmetry should be raised if the imaginary contribution of low energy phase is decreased as the y approaches 0 or n .In all aspects of the prediction of YB, NH 2-a and NH 2-b are quite similar to each other except an overall factor. The YB for NH 2-b is enhanced from both the enhancement of CP asymmetry, A1 (b)/A1 (a) ~ 3.6, and the suppression of wash-out effect, ¡c(b)/ic(a) ~ 4.5. The lower bound of M1 with y = n/2 is pulled down to 2.0 x 1011 GeV for NH 2-b, whereas that for NH 2-a is 3.2 x 1012 GeV. Suppressing a certain Yukawa coupling by putting a texture zero can vary the amount of the asymmetry by order of magnitude.

In Fig. 2, we plot YB as a function of the Majorana phase y for IH 1-a. The different curves correspond to M1 = 5.5 x 1013 to 5.5 x 1015 GeV for fixed M2/M1 = 5.AsinNH, we obtain a lower bound on M1 > 5.5 x 1012 GeV for IH. The prediction of YB for IH with the same value of M1 is smaller than that for NH because A1 for IH is proportional to m2 — m\ which corresponds to the solar mass squared difference, while A1 for NH is proportional to m3 — m2 which corresponds to the atmospheric mass squared difference. We expect from Eqs. (25)-(28) that the predictions of YB's for other cases of IH are almost the same as that for IH 1-a because m1 m2/(2m^ + m2) ~ 1/3.

Although the Majorana phase is not detectable through neutrino oscillations, it may affect the amplitude of neutrinoless double beta decay. Thus, one can anticipate that there may exist a correlation between leptogenesis and neutrinoless double beta decay in our scenario where the heavy Majorana neutrino mass matrix and the charged-lepton Yukawa matrix are both diagonal. The neutrinoless double beta decay amplitude is proportional to the effective Majorana mass \{mee)\ which can be written in the form:

where yt are Majorana CP-violating phases. The \{mee)\ depends on the CP phase y only with inverted hierarchy, so that one can draw a simple correlation between leptogenesis and neutrinoless double beta decay only for the

Fig. 2. YB as a function of Majorana CP phase for case IH 1-a, with various values of M1 where M2/M1 = 5.

Fig. 3. Yb as a function of \{mee )\ for case IH 1-a, with various values of M1 where M2/M1 = 5.

particular case. In Fig. 3, we present a correlation between YB and \(mee)\ for IH 1-a. The inputs are taken to be the same as in Fig. 2. As the value of \(mee)\ approaches to that with y = n/2, the asymmetry is enhanced and the bound of M1 becomes lower. The lower bound of M1 as a function of Majorana phase or that of effective Majorana mass is obtained from the current cosmological observation of YB. In Fig. 4, we present a correlation between the lower bound of M1 and \ (mee) \.

We examined the minimal seesaw mechanism of 3 x 2 Dirac matrix by starting our analysis with the masses of light neutrinos with tri/bi-maximal mixing in the basis where the charged-lepton Yukawa matrix and heavy Majorana neutrino mass matrix are diagonal. We found all possible Dirac mass textures which contain one zero entry or two in the matrix and evaluated the corresponding lepton asymmetries. The baryon asymmetry can be presented in terms of low energy observables, where only one Majorana CP phase among them remains yet unknown. The

.6e+14

O 1.2e+14

0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 <mpp> eV

Fig. 4. The lower bound of M1 as a function of \ (mee) \ for case IH 1-a.

numerical work exhibits the dependence of both the size of baryon asymmetry and the lower bound of M1 upon the low energy CP phase to be clued from neutrinoless double beta decay.

After completing this work, we have been noticed that similar analysis for the hierarchical case in supersym-metric seesaw model appeared in Ref. [27].

Acknowledgements

S.K.K. is supported by BK21 program of the Ministry of Education in Korea. S.C. is supported in part by Grant No. R02-2003-000-10050-0 from BRP of the KOSEF and by Brainpool program of the KOFST. K.S. was supported by the Basic Science Research Institute Special Program of Chung-Ang University in 2004.

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