Doi, Kotani, Takasugi Majorons

8/19/2019 Doi, Kotani, Takasugi Majorons

1/15

PHYSICAL

REVIEW

D

VOLUME

37,

NUMBER 9

1

MAY

1988

Neutrinoless

double-beta

decay

with

Majoron

emission

M. Doi

Osaka

University

of

Pharmaceutical

Sciences,

Matsubara

580, Japan

T.

Kotani*

and E.

Takasugi

Institute

of

Physics, College

of

General

Education,

Osaka

University,

Toyonaka

560, Japan

(Received 18

September 1987)

The

neutrinoless

pp

decay

with

Majoron

emission

is

examined in detail

for

the

0+~0+

and

0+

~2+

transitions.

Especially,

the

quantitative analysis

is

made on the correction

term

which

was

not

dealt

with

properly

before. We found that

the

correction term

does

not

change

the

fundamental

characters

of

the

energy

spectra

of emitted electrons and

the

angular

correlation

obtained

from

the

main term.

The

dominant contribution comes from the main term

through

two

light

neutrino

prop-

agations, except

some

special

cases

which have the

large

mixings between

light

and

heavy

neutrinos.

Also

it

is

shown that

the

0+

~2+

transition

occurs

only through

the correction term,

and it is

by

at

least

—

10

'

smaller than the

0+

~0+

transition.

I.

INTRODUCTION

Recently,

the Battelle

—

South

Carolina

(BS)

group

re-

ported'

the

observation of

the

possible

Majoron-emit-

ting

PP

decay

of Ge.

The

reported

half-life

was

To„ts(0+

~0+

)

=

(621)

X

10

yr.

This aroused

wide-

spread

interest,

because

the detection

of this mode

means

the

direct observation of

the

Nambu-Goldstone

boson

and

the

Majorana neutrinos.

Subsequently,

the

Caltech-

SIN-Neuchatel

and

Santa

Barbara-LBL

(SBL)

groups

gave

bounds

To

lt(0+~0+)

&

1.

2X10

'

and

1.

4X10

'

yr,

respectively,

for

the

half-life

of the

6Ge

decay,

while

the

Irvine

group

gave

abound

To„tt(0+~0+)&4.

4X10

yr

for

the

Se

decay.

All

of

them

seem

to

exclude

the data

TABLE

I. The data

of

the

half-lives

of

the neutrinoless

double-beta

decay

with

Majoron

emission

for

various nuclei and

the

bounds

on

(gs

).

The

bounds

on

(gs

)

are

obtained

by

using

the nuclear

matrix

elements,

Mor '(1

—

Xr),

in

Refs.

g,

14, 24,

25,

and

26. In order

to

derive

the bounds indicated

by

r

(Ref.

26),

we

adopted

the

values

of

nuclear

matrix elements

explicitly given

in

Ref.

27.

Concerning the

ratio

of half-lives

of

'

Te

to

'

Te obtained

by

the

Heidelberg

group,

its

central

value

contradicts

the

theoretical

estimates of the

ratio

for the

(PP)2„mode

by

Refs.

14,

24,

and

26.

Therefore,

the

uncertainty

of the two

standard

deviation

is taken

into

account,

i.

e.

,

(Rr)

&3.

29X10

.

Since the

ratio

estimated in

Ref.

26

is

still

outside this

extended

limit,

the

bound on

(gs)

is

zero

if

both

results are

taken

seriously.

RT

is

the

ratio of

half-lives of

' Te

to

Te.

766e

82Se

100Mo

Experimental

group

Osaka'

Moscow (ITEP)

Batelle-South

Carolina'

Caltech-SIN-Neuchatel

Santa

Barbara-LBL'

Ir

vine'

Osaka~

Ir

vine

Half-life

(yr)

&2X10

&2X10

(6+1)

X

10

&

12X

10

&

14X

10

&

4.

4X

10

&6X

10

&

7.

5X10'

&(5.

0,

13 ,

15,

34,

100')

&

(5

0 13

15P

34q

1QQ

)

(2.

9,

7.

4 ,

8.

6,

19,

60')

&(2.

0,

5.

3 ,

6.

1,

14,

42')

&

1.

9,

4.

8 ,

5.

6P, 13q,

39')

&

(1.

5,

3.

7 ,

3.

8P,

9.

8q,

33')

&(7.

2,

130')

&(6.

4,

120')

' Te/' Te

Missouri'

Heidelberg'

RT

—

X10

'=(1.

03+1.

13)

X

10

&(0.

44,

2.

4 ,

1.

3')

&(0.

28,

1.

5n)

136X

Milano

Moscow

(INR)'

&

1.

6X

10'

&

1.

0X10

&(15,

110')

&(5.

9,

42')

Nd

'Reference

16.

Reference 17.

'Reference

1.

Reference

2.

'Reference

3.

Reference

4.

Moscow

(INR)'

&

7.

0X

10'

IIReference

18.

Reference

19.

'Reference

20.

'Reference

21.

Reference

22.

'Reference

23.

Reference 24.

Reference

14.

Reference 25.

qReference

8.

'Reference

26.

'Reference

29.

37 2575

1988

The American

Physical

Society


2/15

2576

M.

DOI,

T.

KOTANI, AND

E.

TAKASUGI

37

by

the

BS

group.

So

far,

the Irvine and SBL data

gave

the better

bound on

the

Majoron couplings,

aside from

the one derived

from the

geochemical

data on

the ratio

of

half-lives

of

'

Te

to

'

Te.

Summary

of

the

recent

data

and the bounds

on

the

Majoron couplings

derived

from

them are

given

in Table

I.

In

view

of the

above

situation,

the observation

of

the

Majoron-emitting

pp

decay

(pp)o

s

is

an

open

question.

The

important point

is that the

(pp)o,

s

mode

may

be

measured

whenever the observation

of the

0+

~0+

tran-

sition in

the

neutrinoless

pp

decay

(pp}o„

is

made.

Also,

the

(PP)o„~

mode

becomes

a Possible

background

for

the

measurement of the two

neutrino

emitting

pp

decay

(PP)2„.

To

identify

the

(PP)o

s

mode,

the

electron

kinetic-energy

spectra

and

the

angular

correlation

be-

tween

two

electrons become important.

In

particular„

the

experiments

on

Se

and

'

Xe

by

the

visible method

give

both

informations.

As

for

theoretical

progress,

Georgi,

Glashow, and

Nus-

sinov

made the first

analysis

of

the

(pp)o

s

mode in the

Gelmini-Roncadelli

model and

gave

the

half-life

formula

for

the

0+

—

0+

transition.

Subsequently,

Vergados

made an

investigation

on

the

decay-rate

formula.

How-

ever,

in these

investigations,

the electron wave

function

was

treated as

a

plane

wave

multiplied

by

a

nonrelativis-

tic Fermi factor. The relativistic Coulomb

correction

was

taken into

account

by

Doi,

Kotani,

and

Takasugi,

who

gave

the

half-life

formula.

They

also derived

correc-

tion terms

which were

not discussed

before,

and

gave

a

qualitative

argument

on

them.

In this

paper

we

give

a detailed

analysis

on

the

0+~0+

and

0+~2+

transitions

of

the

(PP)o

s

mode.

We

use

a

general Majoron

7

interaction with massive

Majorana

neutrino

N;

with the

mass

m;:

It

has been discussed that the scalar term

f,

j

is

absent if

the

flavor-conserving

terms are considered.

'

Since

we

would

like to examine the

flavor-changing

terms

also,

we

adopt

the

expression

in

Eq.

(1.

1).

The concrete

expres-

sions for

f)

and

g,

in a

general

gauge

theory

are

given

in

Eq.

(A18)

of

Appendix

A,

where

general

properties

of

them are also discussed.

Our aims in

this

paper

are

as

follows.

(i)

We make

a

quantitative

investigation

of

the correction

term

which

has not been

examined in detail

before.

(ii)

We focus

our

special attention on

the

angular correlation

between

two

electrons.

Our

interest was

motivated

by

the

Se data

obtained

by

the

Irvine

group,

who observed

some

excess

events

near the

maximum

kinetic-energy

release,

but in

these

events, two

electrons tend to

be

emitted in the

same

direction.

Since

the

main

term

shows

the

(1

—

os0)-

type

angular correlation, it is

necessary

to

know whether

there

are

any

sources for

the

(I+cos8)-type

behavior.

For

this

purpose,

the

correction

term

and

also the

contri-

bution from

the

right-handed

interaction

are investigat-

ed.

(iii)

Normalization

of

the

half-life

formula for the

0+~0+

transition

is

carefully

reexamined.

We

careless-

ly

missed

the

d(p,

p2)

factor in the definition of the

phase-space integration

in

Eq.

(5.

2.

4)

of

Ref.

8,

which

caused a

factor-2-smaller result

for

the

integrated

kinematical factor

Gz

and

as a

result for

the

decay

rate.

'

Let us

compare

the

(pp)o„s

mode

with the

(pp}o

mode

which

consists of two

parts,

the neutrino mass

(m„)

and

the

right-handed

(

V+

A

)

parts,

corresponding

to

the

m

and

three-momentum

(q)

terms

of a virtual neutrino

propagator,

respectively.

These

two

parts

in

the

(pp)&&„

mode

have a

possibility

to

contribute

as the same order of

magnitude,

because

we have

no definite

theoretical pre-

dictions

for

the

m„and

V+A parameters.

The main

contribution in

the

V+

A comes

from

the

nucleon-recoil

term,

because the

leading

terms are canceled each

other.

However,

in

the

(pp)o„s

mode, the

leading

terms

contrib-

ute

additively,

which come

from

the

products

of

the

four-momentum

parts

of

virtual neutrino

propagations.

On

the

other

hand,

the

V+

A interaction

is

expected

to

be small,

because it is

given

as

a

product

of

m,

and

q,

as

will be shown

in

Appendix

C

explicitly. Thus,

the

nucleon-recoil

term

appears

as a

minor

correction term.

In

Sec.

II

we

give

a

brief

derivation of the

decay-rate

formula

for

the

contribution

from two

light-neutrino

propagations.

In

Sec.

III

the

contributions

from virtual

heavy

neutrinos are discussed

by

evaluating

neutrino

po-

tentials which

are

necessary

for

investigating

the

sizes of

nuclear

matrix

elements.

In

Sec. IV the

0+~2+

transi-

tion

is

discussed.

Concluding

remarks are

given

in Sec.

V.

II. FORMALISM

In

the framework

of

the

Majoron

couplings in

Eq.

(1.

1)

we shall

give

a

brief

derivation of

the

half-life

formulas of

the

0+

~0+

and

0+

~2+

transitions.

We

start from the

R matrix of

the

0+

~

J+

transition

in

the

(PP)o„s

mode which

is shown in

Fig.

1.

Under

the

closure

approximation,

it

is

given

by

N.

3.

FIG.

1.

Diagram

for

neutrinoless

PP

decay

with

Majoron

emission. The

symbols

e, v„M,

N;(0,

+),

N„and

Nf(Jf

)

indi-

cate electron,

electron-neutrino,

Majoron,

the initial,

intermedi-

ate,

and

final

nuclei, respectively.


3/15

37

NEUTRINOLESS DOUBLE-BETA

DECAY

WITH

MAJORON

EMISSION

2577

RI,

—

—

g

U„U,

f

dxdydz

f

&

2 2

ei

e3

2(21T

}

co;coj.

e

'

&k

IX(z) IO&e

'

&JI+

I

Jg

(x)JI

(y)

I0+&

Xg(x,

e,

)yF(T,

' +T~J

')y

(I

+ys}W

(y

e2}

(2.

1)

where

U„

is

the

mixing

matrix

element

for

the

left-handed

neutrino

v,

L

defined by

v,

L

—

g;

U„N,

L,

e„e2,

and

k are

en-

ergies

of electrons and

Majoron,

and

JI

(x)

is the hadronic current. The

lepton

parts

T,

' '

ar.

e.

T,

' =(f

+g);

[co;( E

++L

)+o3/(K,

+L,

+)],

K

++L;+

E;

+Lj

'r,

=

Im,

'—

j;,

+

2(f+-g};,

M142]I

'

k

—

Im,

'ji, +

,

(f+-g}g

[&1&2]I

1

J

~I

J

I

m,

'f—

;,

+

,

(f

+-g);,

M1

42]IL;

~L, I

m'f—

/+

,

(f

+g-}

J(4'1

&2]]&

&J+-

(2.2)

~he~e

q1

—

(o11,

q1),

q1

—

(

—

3;,

q,

),

q2

—

(o1,

,

q2),

and

q2

—

(

—

co~,

q2)

with

m,

=(q1+m;

)'

and

coj

=(q2

+m2)1/2

an

J,

J

=

—,

[f;J(m;+mj

)

+g;J(m;

—

m,

)

]/m,

,

K;~

[a);+pm—

—

,

+

—,

(e,

—

2)k —,

k]

L;+

[o1;+p—

—

,

—

,

(e1

—

2)k

—,

k]

(2.3)

HereP

m,

=

&

E,

&

—

M,

+

M/)

/2, &

E,

&

being

some

aver-

age

of

energies

of the intermediate

nuclear

states,

and

M;

and

Mj

being

masses of initial

and final

nuclei. The

decomposition

into

two

terms

T

and

T

j

'

has

been

done so that the

first

term

T,

.

'

gives

the

main result

which

behaves

like

the

(PP)o

mode, see

Eq.

(B2)

of Ap-

pendix

B.

Therefore,

the

second term

T.

'

will

be

re-

ferred

to as

the

correction

term. The first

term T

is the

one

which

was

considered in

previous

works,

'

while

the second term

Tj

is investigated

quantitatively

in

this

paper.

It is

worthwhile

to

note that

the

leading

contribu-

tion

to the

0+~2+

transition occurs

only

through

the

[g;,

gj

]

part

of T

',

because

of the conservation

of

the

an-

gular

momentum

and

parity,

as discussed in

the

(PP)o„

mode.

'

In

order to obtain

the

decay-rate

formulas for the

0+~0+

and

0+~2+

transitions,

we

use the

following

approximations.

(i)

We

use the

nonrelativistic

impulse

approximation

for

the

hadronic current

with nucleon

recoil

terms

defined in

Eq.

(B4)

of

Appendix

B.

If

the mass

of

inter-

mediate

neutrino

is

heavy,

the

nucleon

form

factor

is

tak-

en into account.

(ii)

We

take the S

and

P waves

for

electrons and

keep

only

up

to

the

first-order

terms

with

respect

to the

P-

wave

and

the nucleon recoil terms.

(iii)

We

take

exp(

—

k

g

R+

)

=

1,

where

k

g

=

(

q2

—

q1)

and

R+

—

(x+y)/2,

see

Eq.

(B9)

of

Appendix

B.

This is

because k

—

MeV

and

thus k

I

R+

I

«1.

Then,

only

the

Swave

of the

Majoron

contributes.

(iv)

We

assume

that

all

nuclear

matrix

elements

are

real.

The

detailed derivation

of the transition formula

within

these

approximations

will be

given

in

Appendix

B.

The

0+

~2+

transition

will be

summarized

in

Sec.

IV.

Below,

we present

the

0+~0+

transition formula

by

assuming

that the

light

intermediate neutrinos

give

the

dominant

contribution. In the

next

section

we shall

dis-

cuss the

contribution from

heavy

neutrinos. The

0+~0+

transition

formula

due

to light-neutrino

propa-

gations

is

given

by

[TovB(0

~0'}]

'=(

I

&gB

&(1

~F1} &f

&(~GT2 ~F2}

I

GBo(1)

—

Re[[&gB

&(1

—

XF1)

&

f

&(~GT2

~F2)]&gB

&

~R3IGBO(2)

—Re[[&gB

&(1

—

&F1)

—

I

&(XGT2

—

XF2}]&gB

&'X3]GBo(3)

+

I

&gB

&&R31

GBO(4

}+«{

&gB &&R3&gB

&*&3}GBO(&

}

+

I

&g,

&x,

I

'G„(6}}

~.

;

',

(2.

4)

where

the

nuclear

matrix elements

Mz&

'

and

7's

will

be

defined

later and

&

I&

=

—,

'

g'

U„U

j[f;

(m;+m )

+g,

(m,

—

m.

)

]/m,

.

&g

&=+'(f+g);,

U„U„,

(2.

5a)

(2.

5b)

The

sum

g'

extends

over

only light

neutrinos,

i.e.

,


4/15

2578

M.

DOI,

T.

KOTANI,

AND

E.

TAKASUGI

37

Giio(k)

=,

f

do(k)dQO,

~,

4

(m,

R)

ln2

(2.

6)

m;,

m-

(m„m,

being

the

electron

mass. In

the

previous

papers,

only

the

(ga

&(1

—

X„,

)

part

of

the first line in

Eq.

(2.

4}

was

considered. This

part

comes from

the main

term

Tfi

in

Eq.

(2.

1)

and

all

others

from the

correction

term

T.

2'.

lJ

The

integrated

kinematical factor

Gao(k)

is

defined

by

bo,

Fcl,

a

[(e,

+ez}m,

/(2e,

ez)]F00,

a+

[(eiez+m, )/(2eiez)]F00,

(2.9)

MGT (Of

Ilhi(r.

)«.

o~

}llo+

&,

(2.

10)

where

F

k

F—

—

Z, ei)Fk(Z,

ez),

the

Coulomb

factor

Fk,

(Z,

e)

being

defined in

Eq.

(3.

1.

25)

of Ref.

8.

In

the

expression

(2.

4)

there

are

various

nuclear

matrix

elements which

are

defined

by

dQO„a

m,

—

iezP,

Pzk5(ci+ez+k+Mf

—

M;

}

Xdeidezdk

d(p,

pz),

(2.

7)

~Fo=(«/g~)

(Of

llh

(r,

~)llo,

+&/MGT

'

(a=1,

2,

3),

~GTa

(of

llh.

«.

)(o. cr~

)llo,

+

&/MGT

'

(a

=2,

3),

where

R is

the nuclear

radius,

p&, p2,

and

k

are

the

mo-

menta

of electrons and

Majoron,

and

(p,

.

pz)

is

the cosine

of

opening

angle

between two

electrons.

The normaliza-

tion

ao„

in

Eq.

(2.

6}

is

the

constant

factor

for the

(PP)0„

mode

defined in

Eq.

(3.

5.

17b) of

Ref.

8.

Note

the

normal-

ization

difference

between

our

new

formula

Gao(1)

in

Eq.

(2.

6}

and

the old

one

Gs

in

Eq.

(5.

2.

4)

of

Ref.

8. The

new

formula

in

Eq.

(2.

7)

contains

the

d(pi

pz)

factor

which

gives

rise

to

the

factor-2

larger

result for

Gso(1)

than

Gs

in

Ref. 8.

The

integrand do(k}'s

in

Gso(k)'s

are

defined

by

.

}llo-

&/MGT

'

(2.

11)

Ya„=i

(cr

„C

C„c—

)

XT3

—

Of

lib

3(r„)(r„~

/R

)T„~

llo,

+

&

/MGT

~3

~T3

6~GT3 I~F3

~

where

r„=

l

r„

l

=

l

r„—

l,

r„=r„

/r„,

and

do(1)=bo„do(2)=2(kR

)bo,

,

do(3)

=

—,

(kR

)[gboi

2(m,

R

)a

—

],

do(4)=(kR

)

b

„Od

(O5)=(kR)do(3),

(2.

8)

gv

(o„XD

+D„Xo )

'2

(D„—D

)

do(6)=

—,

(kR

)

[g

boi

—

(m,

R

)a

+4(mes )

a+],

T„=(r„.

„)(r„o

)

—

,

(o

„cr

)

.

(2.12)

with

(=3aZ+(ei+ez)R,

a and

Z

being

the

fine-

structure constant

and

the

atomic

number

of

the

daughter

nucleus,

respectively.

The

factors

bo&,

a,

and

a+

are

due to the

Coulomb

wave

functions

of two

elec-

trons

and

are

given

in

Eqs.

(3.

5.

18)

and

(C.3.

12)

of

Ref.

8.

As

a

good approximation in the

practical

momentum

re-

gion,

say

p,

)

0.

5m„we

have

Here

the

term

Ya„comes

from the

nucleon

recoil

term,

and the factors

C„and

D„are

defined

in

Eq.

(3.

1.

17) of

Ref.

8. In the

above

expression

we

use the abbreviation

for

the

reduced

nuclear

matrix

elements

which

are

defined

in

Eq.

(B.l.

6)

of Ref. 8.

Three

neutrino

potentials

h

&

h

2

and h

3

are defined

by

m,

R

dQ

1

(E;+E

)(c0;+aij)

hz(r)=

dqe

qr

+K;E

2H

4~

c0;aii

(c0,

.

+co&

)

—

dQ

K;+KJ

co;+coj

h3(r)=

dq

e'q'

2

2+

dr

4ir

N;col

(~.

+ai.

)

(2.

13)

co;=[(q

—

,

kQ)

+m;

]'

co,

=[(q+

—,

kQ)

+m

]'

K,

=(co,

+Pm,

)

(2.

14)

and

virtual-neutrino

energies

and

energy

denominator

due to the

second-order

perturbation are

In

the

above

expression of

potentials,

the

nucleon

form

factor

is

neglected,

because

we are

considering

only

light

neutrinos. In

the

next

section

we shall

take into

account

the

contribution

due

to

virtual

heavy

neutrinos

and

there-

by

include

the form

factor

into

the

integrand of

potentials

in

Eq.

(2.

13).

In

the

N

rnechanisrn,

the

0+

~0+

transition

is

suppressed

from

the similar reason

to

the

(/3P)0

mode.

'


5/15

37

NEUTRINOLESS

DOUBLE-BETA

DECAY

WITH MAJORON

EMISSION

2579

III.

NUMERICAL

ESTIMATIONS

OF VARIOUS

TERMS

IN THE

0+

~

0+

TRANSITION

The

decay

formula

in

Eq.

(2.

4)

has been

derived

by

as-

suming

that

the

light

intermediate

neutrinos

give

the

dominant

contribution.

Here we

shall include

the

contri-

bution from the

heavy

virtual

neutrinos,

and also

evalu-

ate

the

correction terms.

At

first

we

focus our

attention

to the

hierarchical

structure

of

the kinematical

factors

dp(k)'s in

Eq.

(2.

8).

Since energies

of

electrons

and

Majoron

e&, e2,

and

k

are

quantities

of order

of

a

few

MeV,

eR

or

kR is

a

small

quantity

of order

of

(I

MeV)

(5

fm)

—

—

.

Therefore,

for

example,

in

the case

of

Se

with

k

=1

MeV,

dp(2)

and

dp(3)

are about

2(kR)-

—,

'

and

2aZ(kR)-

—,

,

smaller

than

dp(1),

respectively,

and

dp(4), dp(5),

and

dp(6)

have

the

suppression

factor

(kR)

-(~~) in comparison

with

dp(1).

These

hierarchical

structures

are

also

confirmed

from

the

numerically integrated

values

of

Gyp(k

}

for

vari-

ous nuclei

given

in Table

II.

By

combining

these

con-

siderations with the fact that

X&2

and

X3

in

Eq.

(2.

4}

are

expected

to be less

than

1

which

will

be

confirmed later,

Gap(4) Gap(5)

and

Gyp(6)

terms

give

only

very

small

contributions and

will not be

considered.

Although

terms

with

Gsp(2)

and

Gsp(3)

are

expected

to

give

small

correction

we

shall confirm

it at the end

of this

section.

First we

shall discuss the

term

accompanied

by Gzp(1)

in

Eq.

(2.4):

[(gs

~(1

~F1}

(f

~(~GT2

~F2))MGT

(3.

1)

In

order

to

estimate

the

contribution

from

heavy

inter-

mediate

neutrinos

appearing

in the neutrino

potentials

h

&

and

h2,

the

sum of

species

of neutrinos in

(gs

)

and

(

f

)

of

Eq.

(2.

5) should

extend over

the

nuclear

matrix

ele-

ments

MGz

',

X„,

,

XG&2,

and

X„z.

The

potentials

h,

and

h2

are made

out

of the

propagations

of two

neutrinos

with

the

masses

m,

and m as seen

in

Fig.

1 and thus

they

depend

on

m;

and

m

.

In the

case

where the

heavy

neu-

trinos contribute,

say

m

»M

(=

nucleon

mass),

the

nu-

cleon

vertex

should be modified,

because of the

large

nu-

cleon

recoil

effect.

It

may

be

necessary

to

replace

the

nonrelativistic

treatment

of

nucleon

by

the

quark

model.

In

this

paper

in order

to

get

some

rough

idea

on

the

con-

tribution

of

heavy

neutrinos, according

to

Vergados'

and

Haxton

and

Stephenson,

'

we introduce

the

dipole-type

form factor

F(q')

™~

(q'+M~

}'

(3.

2)

with M„=0.

9 GeV.

That

is,

the integrand

of the

neutri-

no

potential

in

Eqs.

(2.13)

is

multiplied

by

F(q2}

for each

heavy

neutrino.

This

modification is not needed

for

the

light

neutrino.

Now,

three

following

typical

cases

will

be

examined:

(a)

m;,

m~

&&pm,

-20

MeV,

(b)

m; &&Pm,

and

m.

»M„,

and

(c)

m~

& m;

&&M„.

Namely,

one and

two form

factors

are

taken into

account for

cases

(b)

and

(c),

respectively.

Explicitly,

in

the limit

of

k

~0

the

potential

h

&

behaves as

0

8

'a

V

eg

0

05

~

~

Ck

CJ

F4

8

O

O

cj

O

V

~

t+f

4P

V

C}

yg

M

I

O

g

~

W

V

05

CP

~

&

g

8~

gg

c5

tQ

e5

a.

&

ce

4P

c

05

o

Cl

og

0

t+I

+

O

I I

I I I

I

OOOOOO

XXXXXX

WmWOnlO

m

M

e4

m

M

W

O~ool

I

O

X

I I

O

O

X X

OO

I I

I

OOO

XXX

OO

Wt

oo

I

I

O O

X X

OO

vl

O

I I

O

O

X X

Ch

O

Ch

I

I

O O

X X

O

t

I I I I I

OOOOOO

XXXXXX

OO

ch~&WWW

I

I I I

OOOO

XXXX

Ch

Q

I I

O O

X

X

Ch

OO

O

OO

I I

O

O

X

X

RS

I I

O O

X X

Ch

rt

I I

O O

X

X

Ch Ch

I

O

X

I

O

X

I

O

X

I

O

X

O

I

O

X

OO

cV

I

O

X

OO

OO

I

O

X

tv'

I

O

X

I

X

I I

O O

X X

™g

I

O

X

c

O

0 0 0

O

O

I

O

X

OO

+

O

X

OO

O

O

X

X

O

O

X

I

X

O

C4

CV

o

Q

+A

OO

cO


6/15

2580

M.

DOI,

T.

KOTANI,

AND E.TAKASUGI

37

(R

/r

)P(x

)

for

case

(a),

h,

=

.

(R/2r)P(x)

for

case

(b),

(M„R/96)[(M~

/m, )

+(M„lmj

)

)J(M„r)

for

case

(c),

(3.

3)

where x

=pm,

r

and

2

P(x)=

—

sin(x)

ci(x)

—

os(x)

si(x)],

J(x)=(x

+3x+3)e

(3.

4)

The

other

potential h

2

in the limit

of

k~0

is

(m,

R

/2IT)[P'(x)+(2/n

)Ko(m,

r

)]

for

case

(a),

h2-

(Rlr)(m,

/mj)

P(x)

for

case

(b),

(MzR

/48)(m,

/Mz

)

(M~z

/m;m )

J(M&

r

)

for

case

(c),

(3.

5)

gLL

U

L

((R Ir)P

)

for

case

(a),

,

gLH

U,

L

U,

H(—(R

lr)P)

for case

(b),

(3.

6a)

(3.6b)

gHHU,

H(M„lmH)

(M„R

/48)(J)

for

case

(c),

(3.

6c)

where L and H

stand

for

the

light

and

heavy

left-handed

neutrinos,

respectively,

(f

+g)LL~H~

is abbreviated as

gLL

~H&

and the reduced

nuclear

matrix

element

(X

)

means

(Of+~)X(o

„a

)~[0~+

).

In

order

to

compare

these

three

types

of

contributions

we

need

to know the

coupling

strength

of

Majoron

to

neutrinos,

gLL,

gLH,

and

gHH.

From

Eqs.

(2.

5.

a)

and

(A18),

we find

gLH

-

mH

ULH

IU,

gHH

-

mH

Iv,

(3.7)

where

ULL

—

UHH

—

is

assumed.

Here

the

parameter

U

where

P'(x)=dP(x}/dx. For the

nonzero

k,

the

analyti-

cal

expressions

for

them are

complicated. However,

the

numerical

integration

shows

that

they

are

essentially

in-

dependent

of

k.

That

is,

for the

variation

of k

from

0 to

2.

5m„h,

is

almost

constant

and

h2

varies

about

5%.

In

Figs. 2(a}

and

2(b),

these

analytic expressions

at k

=0

(the

dashed

lines) are

compared

with the

numerically

integrat-

ed ones at

k=1.

5m,

(the solid

lines).

For

h,

with

m;

=

mj

—

10

m„

there

are

no

visible

distinctions,

so

that

only

the solid lines

are

shown.

Note

that for

case

(c),

the

approximation

k

=0

is

good,

because

m;,

m~

&&k.

Although

the

case of

'

Mo is

shown,

the

approximation

is

equally

valid for

other nuclei.

Now

let

us

examine the main term

(

gz

)

M

G

T

',

the

first

term

of

Eq.

(3.

1),

which consists

of three

types

of

contributions.

corresponding

to

cases

(a), (b),

and

(c).

We

find from

Eqs.

(2.

5),

(2.

10),

and

(3.3)

that these

three

types

are

means

the

energy

scale

of

the lepton-number

violation

defined below

Eq.

(Al 1).

By

combining

Eq.

(3.

6)

with

the

above

estimates,

we find

that

case

(b)/case

(a)-mHUeH/mL

.

(3.

8)

(

j

)XGT2MGT

=

ggLH

UeL

UeH((R

/r)4

)

(3.

9)

It is

amazing

to see

that

the

correction

term

in

Eq.

(3.9)

exactly

cancels with

Eq.

(3.

6b). For

the

Fermi

term,

X„,

and

X„z

in

Eq.

(3.

1),

this

cancellation also

occurs. This

means that the

contribution from one

light

and one

heavy


is

suppressed.

The

contribution

from

case

(c)

is

generally suppressed

by

the factor

0.

1(M„/mLmH )U,

H

in

comparison

with case

(a),

if

it

is

assumed

that

UeL

—

and

(

(R /r

)P

)

—

10(M

z

R

/

48

)

(

J

)

(Ref. 14). In

this

case

(c),

the

contribution

from

the

correction term

(

f

)XGTzMGT

'

does not cancel the

main

term in

Eq.

(3.

6c) in

contrast

with case

(b),

though both

of them are the

same

order.

Therefore,

if

~

U,

2H

~

&10(mLmHIM„),

the

contribution of case

(c)

can be

neglected.

Finally,

let us consider

case

(a)

in

the

correc-

tion

term

of

Eq.

(3.

1). Since it is

accompanied

by

a factor

(f)

~(mL/m,

)

&&1,

its

contribution

is

negligible.

In

summary,

as

far

as

terms

involving

G~o(1)

in

Eq.

(3.

1)

are

concerned,

the

contribution from two

light

intermediate

neutrinos

[case

(a)]

is

dominant

for

~

U,

H

~

&

10(mi

mH

/M„)

which is

satisfied

for most

models.

It

is

worthwhile

to

compare

Eq.

(3.6)

with

the mass

part

of the

(PP)0

mode

given

in

Eqs.

(3.

5.

1)

and

(3.

5.10}

of

Ref.

8,

namely,

This

means

that

if the

mixing

is

not

very

small,

say

~

U,

H

~

&

(mL

/mH

),

case

(b)

could

give

a

sizable

contri-

bution. This

is due to

the

potential

h,

whose values for

case

(b)

is

—,

'

of

case

(a).

In connection

with

this,

we

shall

examine the

correction

term

(

f

)XGTzMGT

'

in

Eq.

(3.

1).

For

case

(b}

it

behaves

as

mL

U2L((R

Ir)p)

for

light

neutrino,

m

M' '='

mH

UzH(M„/mH

)2(M„R

/48}(

J

)

for

heavy

neutrino

.

(3.

10a)

(3.

10b}


7/15

37

NEUTRINOLESS

DOUBLE-BETA DECAY

WITH

MAJORON

EMISSION 2581

12

10

100

0

(a)

m.

=

m.

«um

case

1.

e

m-

=

m.

=

10

ill

-5

3.

e

p

=

20

k

=

1.

5m

e

l

l

l

I

4

l

I

I

l

1

I

l

1

0.4

0.

8

1.

2

1.6

0.

4

0.

8

1.

2

1.

6

FIG. 2.

The

r

de

end

p

dence

of neutrino

potentials

h

&,

h2,

and

h3

for

cases

(a)

and

(b).

The dashe

1

d

k=0

d

h

1'dl'

,

an

t

e so

i

ines

represent

the numerically

integrated

ones

with k

=1.

5m .

For cas

no

visible

differences between

the

dashed

lines and

the

1'd

1'

f

h

i

=

.

m,

. or case

(a),

there

are

lines and solid

lines

agree

with each

other i

d

.

Th'

s

an

e so

i

ines

or

h&

and

h3, so that

onl

the

solid

lines

r

' '

are

shown.

The

dashed

the

analytic

expressions at

k

=0

b

d

h

er

in a

goo

accuracy.

This

shows that otentials

have

ver

can

e

use

as

t

e

approximate

formulas for 6nite

k.

Th

p

very

weak

dependence

on

k

and thus

By

comparing

Eq.

(3.6a)

with

Eq.

(3.

10a),

we find

~GT

™GT

OvB)

(Ov)

(3.

11)

if

all

neutrino

masses are

small

as case

(a}.

Similarly

in

this

case

(a),

X„i

in the

(pp)o

z

mode is

equal

to

XF

in

the

pp)o„mode

defined

in

Eq.

(3.

5.

2)

of Ref. 8. Also

note

that

the

effective

Majoron

coupling

(gz

&

in

the

(pp)o

z

mode is related

to

the

effective neutrino

mass

in

the

(PP)o,

mode

as

Xg

3

or

X3

in

Eq.

(2.

1

1

}.

The

nuclear

parameter

Xz

3

in-

cludes the nucleon recoil

term

Yz„

in

Eq.

(2.12)

which

gives

one more

neutrino momentum

q

effectively.

Thus,

as

it

will

be

shown

later,

(

X&3

~

is

sinaller

than

~

X3

(

in

general,

though

Gso(2}

is a little

larger

than

Gso(3).

Therefore,

let

us

start

to

compare

Eq.

(3.1)

'th

(OvB)

w1

(

gg

&X3MQ'i

Now we

have

a new potential

h

3

which

behaves

in

the

k

~0

limit

as

(g~

&-(m„&/U,

(3.

12)

1,

2

2x

P(x

)

——

P'(x

)+

—

for case

(a)

m'

where

Eqs.

(3.

6a),

(3.

7),

and

(3.10a)

have

been used.

Next, we shall

examine

the

terms

associating

with

G~o(2)

and

Gzo(3)

in

the second

and

third lines

of

Eq.

&

.

,

,

which

give

the different

electron

energy

spectra

and

2.

4&

angular

correlation.

At first

we notice

that

these

terms

are

the

interference

of

the

factor

in

Eq.

(3.

1)

with either

1

,

(pm,

/m

.

)

—[P(x

—

P'(x

)

]

for

case

(b),

(3.

13)

,

,

(M„/m;m.

)

—[(M„r)+(M„r)]e

for

case

(c),


8/15

2582

M.

DOI,

T.

KOTANI,

AND E.TAKASUGI

37

where

M

is

the

nucleon

mass,

P„,

P,

and

q

are

momenta

of

initial

nucleons and a

virtual

neutrino,

respectively,

and

fR

=-',

(gv/g~

)f

w

,

,

(gp/gg

}(E„+E

———

—„'

E'

)—

ith

the

strength

of

the

weak magnetism

f

z„

the

induced

pseudoscalar

form factor

gp,

energies

of

initial

and

final

nucleons

(E„,

E

)

and

(E„',

E'

),

respectively

Further.

-

more,

we

keep only

the

q.

r„

term

because

it

has a

large

coefficient

fR

—

,

.

Note

that

q

is

a

difFerential

operator

—

B/Br„.

Under these assumptions

we

find

~R3

—

&0f

llfRh3R(r.

)(~.

~

}ll0

&/MoT~

where

(3.

15}

1 1

d

2

1

r

2M 2M

r

dr Mr

2

(3.16)

For the

light-neutrino

case

(a),

h3R

-h3(R)/(MR)-

where x

=Pm,

r

and

(()

is

defined in

Eq.

(3.

4).

In

case

(a),

as seen from

Fig. 2(a),

h3

is almost

independent

of

r,

so

that

73

is an

order of

or

less

than 0.1.

By

taking

G&0(3)/GR0(1)

—

,

,

into

consideration,

we can

safely

neglect

the correction

term

of

G&0(3)

for

this

case

(a).

Within this correction term

of

(gz

)X3MoT

',

let us

com-

pare

the

other two

cases

(b)

and

(c)

with

case

(a).

Con-

cerning

case

(b),

the

potential

h3

takes the

value

about

(M„ /mH )

X

10,

as

seen

from

Fig.

2(b).

Then

the

ratio

of

case

(b)

to case

(a)

becomes

roughly

[gL~U,

~{Mg

/m~)

X

10

]/(0.

2gLL

)

—

U,

H(Mg/mLm~),

~

Therefore,

case

(b)

can be

safely neglected

under

the

con-

dition

U,

~

~

10(mz

mH

/M„).

As

for

the

final

(c)

case,

h3

is

smaller

by

the

factor

(M„/rnH)

than

h,

for case

(a)

and then this

combination

due to

two

heavy

intermediate

neutrinos can be

ignored.

In

summary,

the

correction

term

accompanied

with

GR0(3)

in

Eq.

(2.

4)

can be

neglected as

the

first

approximation.

The

remaining problem

is

to show that

l

X3

l

l

XR3

and therefore

the correction term with

GR0(2)

can also

be

ignored.

In order

to evaluate

XR3MGT

'

in

Eq.

(2.

11),

we

decompose

the

Yz„r„

term

into the

irreducible

ten-

sors with

respect

to the

spin

space.

Since

this

term

ap-

pears

as

a

correction term

we

shall take

only

the

leading

(o„o

}

term,

because

the

main contribution

is

con-

sidered

to

be

a spin-singlet

state of

two

nucleons. The

decomposition

is

very

similar

to

the

(pp)0„mode

case. In

the

approximation

where we

neglect

momenta

of

elec-

trons

p„pz,

and

Majoron

k

in

comparison

with

those

of

nucleons,

we

find

2M(YR„~

r„~

)=i(a

„n~

}[fR(q r„~

)

—

—

,

(P„—

)

r„],

(3.

14)

0.

3)(4&(

10

—

10,

and we

find

XR3-

—

,

X

10

/0.

7-

210

.

(3.17)

Therefore,

we

can

say

that

lXR3

l

&

l

+3

l

-0.

1.

Thus, we

conclude

that the

contribution

from

the

light

neutrino

propagations

in

the main term

is

dominant,

ex-

cept

the

special

model

with

l

U,

zz

l

&10(ml

m~/M„)

.

Therefore,

the

approximation

to

keep

only

the

first

term

of

Eq.

(3.

1)

is

valid.

IV. THE

0+—

2+

TRANSITION

The

0+

~2+

transition

becomes

possible

only

through

the

[g;,

g ]

part

of the

correction term

Tf;

'

in

Eq.

(2.

2};

namely,

the

term

T&,

which

is

a main term for the

0+~0+

transition

does not contribute.

Thus the

half-

life is

expected

to be

very long.

The

half-life

forinula in

the 2n

mechanism is

given

by

[T0„R(0+~2+)]

=

I

(gR

)NR21

(4.

1)

Here

(gz)

is

defined

in

Eq.

(2.

5a) and

the

integrated

kinematical factor

GR2

is

defined

similarly

to

Eq.

(2.6}

as

with

the

definitions

of

b21

and

b22

in

Eq.

(C.

4.

12)

of Ref.

8. As

a

good

approximation

in

the

practical momentum

region, say

pj.

&

0.

5m„b2&+b22

becomes

'2

2

1 P& p2

{b21+b22)

F10+

F01

3

@le

m,

where

F~k

is defined

below

Eq.

(2.

9).

The

nuclear

matrix

element

NR2

is defined as

NR2

(2f+

llh

3(r„—

(r„

/R

)

Y„'

r„r„

ll0,

+

),

(4.

4)

where

—

,

(gr/gg

)'(3'

+

—

gr/g„)e'

(o'„+0'

)

.

(4.

5)

At

first

sight

we

may

notice

that in

the integrated

kinematical

factor

Gzz,

there

is a

suppression

factor

(kR)

(m,

R

)

-(

~1

)

( —,

,

)

—

10

in

comparison with

G&0{1)

in

Eq.

(2.

6).

Because of

this,

the

0+

~2+

transi-

tion

is

highly

suppressed.

The

numerical

value

of

G~2

is

given

in Table

II

also.

Since

the neutrino

potential

h

3

ap-

pearing

in

NR2

of

Eq.

(4.4)

is

much

smaller than

h,

in

MGT

'

as

shown in

Fig.

2

and

Eq.

(3.

13),

the

half-life

of

the

0+~2+

transition

is

at

least

10

times

longer

than

that of

the

0+

~0+

transition.

In

the

N'

mechanism,

the

half-life formula for

the

0+

—

2+

transition

is evaluated

similarly

to

the

(pp)0„

mode.

The result

is

[TNe

{0+

2+)]

1

=

l

&gR

&Na

l

a(a)

l

&

ef

l

e;

&

l

GR2',

(4 6)

G&2

—

2

J

(kR)

(m,

R

)

(b21+b22)dQ0

B

4

(m,

R)

ln2

(4.

2)


9/15

37

NEUTRINOLESS

DOUBLE-BETA

DECAY

%'ITH

MAJORON EMISSION

2583

&a

—

p

i

—,

h3(r„)(r„

/a)

i

&

)

=&3(o),

'2

4

80

a

1

B2

9

g

B2

&

(4.7)

(4.

8)

V.

CONCLUDING REMARKS

In our

previous

paper

we

have derived

the

correction

term

for

the

Majoron emitting

pp

decay,

and

gave

only

a

qualitative

argument

on

it.

In this

paper

we investigated

this correction term

quantitatively with

the care

of the

effect

ofmasses of intermediate neutrinos.

We considered

three cases:

two

neutrinos

are

light [case

(a}],

one is

light

and

the other is

heavy [case

(b)],

and

two

are

heavy [case

(c)].

We found

that there is

a

cancellation between

the

main

and

the correction

terms

for case

(b),

although

each

term for case

(b)

itself could

be comparable to the

main

term

of case

(a).

All other

correction

terms

turn out to be

small

and also the main term

for

case

(c)

is smaller

than

that

for

case

(a)

for

i

U,

&

~

&

10(mL

mz/M&

)

which

is

satisfied

in

most models.

Thus,

we concluded

that

the

decay-rate

formula is

given by

the

main

term

for

two

light


[case

(a)]:

[TO

B(0+~0+)]

=

I

(ga

&

I

'

I

MGT

I

'

I

I

&F

I

'Geo(1)—

(5.

1)

where

G~o(1)

is

defined

in

Eq.

(2.6) and

MGr' and

XF

are

nuclear

matrix elements

which

appear

in

the

(PP)o

mode

[see Eqs.

(3.

5.

1)

and

(3.

5.

2)

in

Ref.

8],

because

M~Gr

'

=M/~

as

shown

in

Eq.

(3.

11).

Once

again

we

re-

mind

the reader that

the

value

of

Gzo(1)

is twice

as

large

as

our old definition of

Gz

in

Eq.

(5.

2.

4)

of Ref. 8.

The

single-electron

kinetic-energy spectrum

and the

sum-energy spectrum

were

obtained

by

using

Eq.

(5.1)

and

the result was shown in

Figs.

6.

10 and

6.

11

in

Ref. 8.

In

our

recent

paper'

we

discussed some

approximation

schemes

and

compared

them

for

the

(PP)z„, (PP)0„,

and

(PP)0,

~

modes.

One

of our

concerns for

the correction

term

T&,

'

in

Eq.

(2.

1)

was to

investigate

the

possible source for

the

(1+cos8)-type

angular

correlation

between

two

elec-

trons, while the

main term

Tf';

in

Eq.

(2.1)

shows

the

(1

—

os8}-type behavior. With

this

in

mind,

we

exam-

ined

the

correction

term which

is

given

in

Eqs.

(B10)

—

B12}. From

this formula

we

find

that terms

which

give

the

(1+cos8)-

or

isotropic-type correlation

where

the

parameter

a is the

mean distance

between

quarks

inside

the

hadron

(a

-0.

7

fm),

P (b,

)

is

the

proba-

bility

of

finding

b,

inside

the

nucleus, and

(4/

i

4;

)

is

the

overlap of the

initial

and

final

residual nuclei.

The

con-

tribution

from

the

N*

mechanism

is

given

roughly

by

[~~.

(0+

2+

}]-

—

10

i

h3(a)/&az

i

[7

(0+~2+}]

If

&a&-10

~

as

suggested

from

the

(pp)o„mode,

'

the

N*

mechanism

gives

the

same-order

contribution

as

that

of

the

2n

mechanism, and thus its

contribution

is

negligi-

bly

small.

are

suppressed

at

least

by

a factor

(m,

R)(kR)(10

Also,

they

always

contain

the

term

X3

in

Eq.

(2.

11),

which

is

expected

to

be

of the order of

—,

.

Thus,

the

angular

correlation

is a

(1

—

os8)

type

with a

good

accuracy.

In

the

recent

paper,

'

we

discussed that

the angular

correla-

tion

is

very

well

expressed

as

1

—

(p,

p~/e,

e~)

cos8

.

(5.

2)

APPENDIX

A:

COUPLINGS

OF

MAJORON

WITH

NEUTRINOS

Majoron

is a Nambu-Goldstone

(NG)

boson associated

with the

spontaneous breaking

of

the

global symmetry

for

the lepton-number

conservation. Here

we

examine

a gen-

eral form

of

the

NG-boson

coupling

with neutrinos.

As for

the

charged leptons we use

the

basis where

they

are

in

mass

eigenstates.

Then, we

put

all

neutrinos in

a

single multiplet

VL

c

(va )

(A

1)

Also,

we have

examined

the contributions from

the

right-handed

interaction,

because

the

(1+cos8}-type

an-

gular

correlation

appears

if one

lepton

vertex

is

V

—

A

and the

other

is

V+

A.

In

Appendix

C

we evaluated

the

yield

from

the

right-handed

interaction.

However,

this

contribution turns

out to be

very

small

as

expected.

Haxton

and

Stephenson'

proposed

to

obtain the

upper

bound

of the

combination of

the

heavy-neutrino

mass

(mz)

and

its

mixing

matrix

with the

electron

weak eigen-

state

(U,

~)

from

the

(PP)o„mode,

as

seen

from

Eq.

(3.

10b).

They

assumed that

there is

a positive

interfer-

ence between

contributions

due to

propagations

of the

light and

heavy

left-handed

neutrinos

and

that the

treat-

ment

of the

nonrelativistic

approximation with

the

nu-

cleon

form

factor for

the nucleon

current

is still

applic-

able for

mz ))M„,

M„being

a

parameter

of the nucleon

form

factor

in

Eq.

(3.2).

If

we

accept

similar

assump-

tions, we can derive the

bound

on

g&~(

U,

&

/m&

)

=

U,

~/(Umz)

from the

(PP)0„~

mode as

seen

from

Eqs.

(3.6c)

and

(3.7).

Here

U

is

the vacuum

expectation value

of the

Higgs

boson,

which

leads

to the

spontaneous

breaking of the

lepton-number

violation.

Another

purpose

of this

paper

was to settle

the

normal-

ization

problem

of

the

(PP)o„z

mode. If

the

normaliza-

tion

of

the interaction

in

Eq.

(1.

1)

is

used,

our old

kinematical

factor

Ga

in

Eq.

(2.5.

4)

of Ref. 8

should

be

multiplied by

2

as

given in

Eq.

(2.6).

However,

this

is

still

smaller

by

factor

2 than that

referred

to

by

Avignone et

al.

'

as the

result

by

Georgi

et

al. Because

of the

normali-

zation

problem,

there

was a

confusion on

the

values

of

(g~)

given

by

various

experimental

groups.

Therefore,

we

made

Table I

to summarize

the

experimental data and

the bounds

on

(g~

)

derived from

these data.

Note

added

in

proof. Recently, we received

a

paper3o

by

the LBL-Mount

Holyoke

College-New

Mexico

group

which

reported the

half-life

limit

on

the

neutrino-

less

pp

decay

with

Majoron emission

of

'mMo;

To„a(0+

~0+

}

&

2.

1

X

10

yr

with

90%

confidence.


10/15

2584

M.

DOI,

T.

KOTANI,

AND E.TAKASUGI

37

We

take

all

Higgs

fields to be

real.

Then,

the Yukawa

in-

teraction

is

written

as

(4j

)NG

compo

cot

(0

)

jX/v

(A12)

(A2)

Here

I

P=

I

((}

and

I

j

is

a

matrix

satisfying

I

=I

because

(O',

L,

)

%kt

——

(Cbt

)

%,

L,

.

Now

we

assume

that the

Lagrangian

is invariant under

the

gauge

symmetry

transformation

which conserves the

electric

charge,

By

substituting

Eq.

(A12}

into

(A2),

the NG

boson

cou-

plings

to neutrinos

are found to be

[(4

}

IOu&II

+4

I

tOv(%

)

]X.

(A13}

2v

By

using

the

relations

in

Eq.

(A5),

I

Ou

is

expressed

in

a

much

more convenient form:

5+1

—

it—

,

e,

(x)%t,

5&}}=

—

8,

e,

(x)P,

(A3)

[

{4

I)

(X

m+mX}VL

2v

and a

global-symmetry

transformation

&G+I,

=

iX—

+t

&

&G((}=

i8—

d

&

{A4)

(Asb)

Also,

the

invariance

of

Higgs

potential

Vis

expressed

as

BV BV

{8o

)jk(t

k

0

Ojkek

J

(A6)

In connection

with the spontaneous

symmetry

break-

ing

of

the

gauge

and

global

symmetries,

we

introduce

where

t,

and

X

are

Hermitian

matrices,

8,

and

8

are

pure

imaginary

antisymmetric

matrices,

and

e,

(x

}

and

e

are

infinitesimal local

and

global

parameters.

The

invari-

ance

of

L

„under

these

symmetries

leads to the

identities

(A5a}

++L(Xm +m

X

)(+L

)

]X,

where m

is

the

neutrino

mass

matrix

defined

as m

=I

v

and

(A14)

(A15)

=X

—

g

(Pq/p~)tv

.

B

As

expected,

the NG boson

coupling

to

neutrinos

de-

pends

on neutrino

masses and

the

transformation

proper-

ty

of neutrinos under

the

broken

gauge

and the

global

symmetries,

except

for

normalizations

pB, pB,

and

v.

Let

us

rewrite

Xr

in

Eq.

(A14)

in terms

of

mass eigen-

state

neutrinos

defined

as

l.

—

UNL,

,

(A16)

where N is

the

Majorana

neutrinos

defined as

N

=NL

+ (NL

)

=

N

.

Then,

the

NG

boson

couplings

to

neutrinos

are

Xr=

—

N;(f

j+g

jy—

)N

X,

(A17)

(

j=Pj+vj,

{A7)

with

(((}')

=0, so

that

leptons

acquire

masses and both

gauge

and

global

symmetries

are

broken.

There are

massless

bosons

associated

with

the

symme-

try

breaking.

By

differentiating

Eq.

(A6)

once

with

respect

to

((},

and

putting

&I};

=

u;,

one finds

where

[UtXU

—

UtXU)

]j,

m;+m.

[U

XU+(U XU)r],

2v

(A1Sa)

(A1Sb)

Mj(8,

u

}j

=0,

M2(8u

}

=0,

(AS)

where

M

j=(B

V/BP;BP

)& „

is

the

mass matrix for

Higgs

bosons.

Equation

(AS)

means

that the vectors

(8,

u)

span

the

unphysical

Higgs

bosons which

will

be

eaten

by

gauge

bosons.

The NG boson

associated

with

the

global-symmetry

breaking

must correspond

to

the

vector

Ou

=8v+Xkb8kv

satisfying

u

8,

0v

=0

for

all

a,

where

k&

is

a

constant.

Now

we define

the

orthogonal

basis

of the

space

spanned

by (8,

u)

as

{8~

u):

v

OAOBv

5ABI

A

T

2

Then,

we find

Ou

=

Ou

—

g

(pv /p~

)8v

u

B

(A9)

(A

10)

X=

—

&II'

0

u/u

(Al 1)

where

v=(p

gP~/p~)'~

and

y—

u

08v.

,

By

invert-

ing

this we

find

with

p,

B

——

u

8BOv=v

OOBv.

That

is,

the

NG

boson is

expressed

as

Since

we

started from the basis where

charged leptons

are

in

mass

eigenstates,

U is the neutrino

mixing

matrix.

In

the

previous works,

'

the

second

term on the right-hand

side of

Eq.

(A10) was not

taken

into

account.

This

term

is needed to

guarantee

that the

NG

mode

associated

with

the

global

symmetry

is

orthogonal to

the

unphysical

Higgs

modes from the

local

gauge

symmetry

breaking

as

discussed

just

above

Eq.

(A9).

Let us examine

the

meaning

of

the

general

formula

of

the

NG boson

couplings

with

neutrinos

in

Eq.

(A17).

First,

we

notice that

the flavor-conserving

(diagonal)

cou-

plings

are

always

pseudoscalar

(f;;

=0),

while

flavor

changing

couplings

may

have both scalar

and

pseudosca-

lar

parts.

Next

let us confine

our

discussion to the case

where there are no

right-handed

neutrinos

v&.

If

there

are no

gauge

symmetries connecting

difFerent

generations

(local

horizontal

symmetry),

t~

must

be

flavor

diagonal.

Then,

if the

same

global

charge

is

assigned

to all

genera-

tions,

then

f

J

=0

and

g~.

=

—

(m;

v/)(

.

X);;

;6,

There-

fore,

the

different

global

charges

should be

assigned

to

different generations

to

obtain the

off-diagonal

term

in

g

and

f.

If

va

exists,

the

global

or

local

charges

for

vL

are

different

from

those

for

v&

and

thus

the

ol'-diagonal


11/15

37

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DOUBLE-BETA

DECAY

WITH

MAJORON

EMISSION

2585

terms

will

appear,

but

they

wi11

be

very

small.

The above formula

is derived

for

real

Higgs-boson

fields,

but

it

can be converted into

complex

Higgs-boson

fields

related to

real

fields

as

t

„=

Ts

—

Y/2

=

1

for

vL

.

Also

the

global

charge

X=B

I—

is

0=diag(0,

—

,

0,

—

}

for

4 and

X=

—

for

v~.

Then, noticing V

=(u,

v,

u, v)/+2, we

find

It,

=4v,

pz

—

—

U,

pz

—

+40

.

Now

we find

1

—

:

P=

1 0R

(A19)

X=

[2v(P

P*—

—

(h

—

dP')]

.

(A21)

[2(

2+

4v

2)

]1/2

where

QR

and

pt

are

real

and

imaginary

parts

of

y

and

form the

basis

of

real

representation

(().

Then,

the repre-

sentation

matrices for

complex

fields

are

8„=E&„E

and

8=ESEt.

The NG boson

associated

with

the glo-

bal

symmetry

is

Note that

this

definition

of

7

is

twice

larger

than

that

in

Eq.

(4.10) of Ref. 5.

With

this definition, the

Majoron

coupling

with neutrinos derived from the

Yukawa

in-

teraction in

Eq.

(4.

19)

of

Ref.

5 agrees

with our

definition

in

Eqs.

(1.1}

and

(A17).

The

Majoron

couplings

to

neutri-

nos are

now

given

by

X=1@

S

V/v,

(A20)

J

V

'

gV

v

(

2+4

2)1/2

V

(A22}

with

V=Ev and

S=ESE

.

By

knowing

that

ps

=

Vt8RSV and

)u,

2R

—

V

8RHR

V,

the

NG coupling

is

immediately

derived.

For

example,

let us consider the

Gelmini-Roncadelli

model.

This model includes

the

left-handed neutrino

vL,

a

doublet

Higgs

boson

P,

and a

triplet

Higgs

boson h,

.

In

this

case,

the

global

charge

X

is

B

—

L.

Since the electric

charge

Q=Ts+Y/2

is

unbroken, the

broken

gauge

symmetry generator

will

be

8„=Ts

—

Y/2

=diag(1,

2,

—

1,

—

)

for the

basis

of

4

=(P,

b,

,

P

',

b,

'}

and

APPENDIX

B:

DETAILED FORMULAS

FOR

THE

0+

~0+

AND

0+

—

2+

TRANSITIONS

In this

paper

we

have

adapted a

general

form

of

Majo-

ron

couplings

in

Eq.

(1.

1),

so

that the

derivation

becomes

slightly

different from that in

Ref. 8.

Here we

shall

give

a

brief

explanation

of the

derivation and

show

the result.

We

start from the

S

matrix

derived from the interac-

tion

(1.

1):

d'qi

d

q2

S=

—

—

—

g

U„U,

f

d

x

d

y

d z

f

f

(J+

~

T[Jft(x)JLt(y))

i

0+&e

—

iq]

(x

—

)

e

X

q,

—

m,

.

+iz

—

iq&(z

—

)

(k

~nz)

~0&

',

q2

—

m

+is

Then

we use

the

following

decomposition:

X

(x,

ei)yp(

1

—

ys)( 1+m;

)(

f,

,

+gi&ys}(42+m,

)(1

ys)y

0

(y

e—

2}

(B

1)

(1

—

ys)( 1+m

)(f,

+g;,

ys)(tt2+m,

)(I

—

ys)

=(f

+g),

,

[(q

1

m;

)+(q2

—

m,

')](I

—

ys)

+[(f

+g);,

t [ „ 2]

—

(q,

—

q,

}'j+

f,

,

(m,

+m,

)'

+g

J(m;

—

m

)

](1

—

ys)

.

By

substituting

Eq.

(B2)

into

Eq.

(Bl)

and

making

the

x,

y,

z,

and

then

q,

and

q

2

integrations, we

obtain

the

formulas

given

in

Eqs.

(2.

1)

—

2.

3).

The

first

line of the

right-hand

side

of

Eq.

(B2) correspond

to the

main

term

Tf,

and the

other

two

lines

correspond

to

the

correction

term

Tf;

'.

We

use

the

impulse

approximation [approximation

(i)]

for

the hadronic

current,

Jg

(x)

=g„g

(Gv„g&+G'„„gt

}5(x

—

„),

(B3)

where

Gvn

=~a

[(gv/ga

}

—

C„],

Ga„—

n

[on

(gv/ga

)D'„] .

(B4)

The

operators

C„and

D„are

the

nucleon-recoil

contributions and

are

defined

in

Eq.

(3.

1.

17)

ofRef.

8.

By

performing the

partial-wave

decompositions

of the emitted

Majoron

and

introducing the

form

of hadronic

current

in

Eq.

(B3)we

find


12/15

2586

M.

DOI,

T.

KOTANI, AND

E.TAKASUGI

37

i

G

~z

2

2

1

2mR

v'2k

X(Jf+

~

g

(r„,

e,

)g(2l+1)PI

ik

(X„+Z

}'

X,

)(I+'YsW'

(r

B(kR+„)

(B5)

~

~

A A

where

Q

is

replaced

by

idly(kR+„)

in

the

Legendre

function

Pl(k

Q.

)

which corresponds

to the

i

wave of the

emitted

Majoron.

By

using

f,

in

Eq.

(2.

3),

we defin

X„m

=g

U„UiI[(f

+g}(ih,

f

jh—

](GVnGVm

GAn

GAm

}

+(kR

)(f

+g);,

(h3+

,

h4+—ihS)

[(GVnGAm

G—

AnGVm

)+i

(GAn

&&GA )]

j,

(B6)

X'm=g

U„U„I[(f

+g)~ih,

flh2—

[GvnGAm

—

GAnGvm

—

(GAn

XGAm)']

+(kR

)(f+g)

J(h3+

—,

h4+ih

s

}[(Gvn

Gvm+GAn'GAm

)5

—

GAn

GA +GAn

GA

)

+i&'

«V.

GA

+GA.

Gv

}]j

.

(B7)

The

q

integrations

are

included

in neutrino

potentials

h

„h

2,

hs,

h4,

and

hs

which

are

defined

by

h,

=

—

f

IdqdQj[~,

(K,

++L,

)+co

(K;

+L,

)],

—

4

Br„

m,

R

K;

+L

h2=

'

[dqdQj

cl

tdqdQj

Si

+Sj

Q)i

+coj+k

K,

+Lj

K,

+L,

co

i

+

co

j

—

Q)

i

+

67j

+

k

(K;

K~+

L+LJ

)—

(B8)

f

tdqdQj

(as;

—

,

)

'k

—

'

+(as;+roj)(K;

K)+

L,

+L,

)—

.

coi+coJ

—

co

+Q)

+k

a

K;

+

LJ

Kj++L,

+

X

f

IdqdQj

'

+

'+

'+

+(K,

K.

+L

L.

+nm

;+Nj

—

co;+co +k

where

R+„m

—

(r„+r

)/2

and

dq

iq'r~~

d

—

kg

R

N;

COJ.

4~

(B9)

and

K;~

and

L,

+

are

defined in

Eq.

(2.3).

Now we use the

approximations

(ii), (iii),

and

(iv)

defined in Sec. II.

By

applying

approximation

(iii),

i.e.

,

exp(

ikQ

R+

)

=—

and

neglecting

ej

and

k in

comparison

with

pm,

in

K,

+

and

L;

z,

neutrino

potentials

h,

and

h

take

th««m in

Eq.

(2.13)

and

hs

i

rh3

w—

—

ere

h3

is

defined in

Eq.

(2.

13).

Other potentials vanish,

i.

e.

,

h

=h

=().

(B

1

}

The

0+

~0+

transitions:

ao„

dr,

„,

(0+

0+)=,

'

[A,

+(p

~

p

)B

]dQ

4n.

(m,

R

)

where

ao„and

d

Qo„B

are

defined in

Eqs.

(2.

6)

and

(2.

7),

and

AO=Ibo&

jX+

,

g(kR)Y

j

+

',

(m,

R—)(kR)

f(—

m,

R)a+

—

'a

]

j

Y

j

—

—

,

(m,

R)(kR)a

Re(X'Y)j,

Bo=

—

[bo,

'

~

X+

,

g(kR)Y

~

,

(—m,

R)

—

k—R)

aB

i

Y'~

83(m,

R)(kR—

a—

zIm(X'Y)j

.

(B

1

0)

(B

1

1)

(B12)

Here

bo,

',

az,

and

aI

are

defined

in

Eqs.

(C3.

17) and

(C3.

18) ofRef.

8,

and

by

assuming

that the

light

intermediate

neu-

trinos

give

the dominant

contribution, X

and

Yare

I

~gB

}l(1

XF1)

(kR)XR3]

~

j

}(XGT2 XF2)

j~GT

Y

(gB

}X3~GT

(B13)


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37

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%ITH

MAJORON

EMISSION

Documents

Doi, Kotani, Takasugi Majorons