1 NEUTRINO PHYSICS Anatoly V. Borisov Department of Theoretical Physics, Faculty of Physics, M. V. Lomonosov Moscow State University, Moscow, Russia Lecture AB-I. Properties of neutrinos. Lecture AB-II. Rare meson decays mediated by Majorana neutrinos. Lecture AB-III. Heavy Majorana neutrinos in high-energy lepton-proton and proton-proton collisions.
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1
NE
UT
RIN
O P
HY
SIC
S
An
ato
ly V
. B
ori
sov
Dep
art
men
t of
Th
eore
tica
l P
hy
sics
, F
acu
lty
of
Ph
ysi
cs,
M. V
. L
om
on
oso
v M
osc
ow
Sta
te U
niv
ersi
ty,
Mo
scow
, R
uss
ia
Lec
ture
AB
-I. P
rop
erti
es o
f n
eutr
ino
s.
Lec
ture
AB
-II.
R
are
mes
on
dec
ays
med
iate
d b
y M
ajo
ran
a n
eutr
ino
s.
Lec
ture
AB
-III
. H
eavy M
ajo
ran
a n
eutr
ino
s in
hig
h-e
ner
gy l
epto
n-p
roto
n a
nd
pro
ton
-pro
ton
co
llis
ion
s.
2
Lec
ture
AB
-I
PR
OP
ER
TIE
S O
F
NE
UT
RIN
OS
OU
TL
INE
�
Bri
ef h
isto
ry
� N
eutr
ino
fla
vo
rs
� C
hir
ali
ty a
nd
hel
icit
y
� O
bse
rva
tio
n o
f n
eutr
ino
osc
illa
tio
ns:
no
nze
ro n
eutr
ino
mass
es a
nd
m
ixin
g
� L
imit
s o
n t
he
neu
trin
o m
ass
es
� T
he
natu
re o
f n
eutr
ino m
ass
es:
Dir
ac
or
Ma
jora
na?
� T
he
“se
esaw
” m
ech
an
ism
for
neu
trin
o m
ass
es
� C
on
clu
sio
n
3
Bri
ef h
isto
ry
T
he
firs
t ev
iden
ce f
or
the
neu
trin
os
ap
pea
red
in
th
e st
ud
y o
f n
ucl
ear
bet
a
dec
ay
s (1
92
0’s
): (
,)
(,
1)?(
)→
±+
+�
AZ
AZ
en
oth
ing
else
visi
ble
wit
h a
con
tin
uu
m �
e s
pec
tru
m r
an
gin
g f
rom
e
m u
p t
o
max
.≡
−�
ei
fE
QM
M
N
iels
Bo
hr
spec
ula
ted
ab
ou
t th
e p
oss
ibil
ity o
f en
ergy n
on
con
serv
ati
on
.
I
n D
ecem
ber
19
30 W
olf
gan
g P
au
li s
ugg
este
d t
he
exis
ten
ce o
f a u
nob
serv
ed
neu
tral
an
d v
ery
lig
ht
pa
rtic
le w
ith
sp
in ½
ta
kin
g i
nto
acc
ou
nt
con
serv
ati
on
of
elec
tric
ch
arg
e, e
ner
gy
(m
ax0
−�
eQ
E)
an
d a
ngu
lar
mom
entu
m (
an
d
sta
tist
ics)
. T
his
new
part
icle
wa
s n
am
ed n
eutr
ino (
“sm
all
neu
tro
n”
in
Ita
lia
n)
by
En
rico
Fer
mi
in 1
932
.
I
n 1
934
E. F
erm
i p
rop
ose
d t
he
firs
t th
eory
of
bet
a d
eca
y a
nd
wea
k
inte
ract
ion
s in
gen
era
l. T
he
inte
ract
ion
L
ag
ran
gia
n o
f F
erm
i’s
theo
ry
2Fw
eak
GL
JJ
µ µµµµ µµµ+
=−
is t
he
pro
du
ct o
f tw
o w
eak
cu
rren
ts a
t th
e sa
me
spa
ce-t
ime
poin
t, i
. e.
, 4
-
ferm
ion
in
tera
ctio
n w
ith
zer
o r
ad
ius
(in
fa
ct, ver
y s
hort
ra
ng
ed).
4
Th
e co
up
lin
g s
tren
gth
(F
erm
i co
nst
an
t)
5
52
2
10
1.1
6637(1
)10
GeV
.�
F
p
Gm
− −−−−
−−
−−
−−
−=
×=
×=
×=
×
Fer
mi’
s th
eory
pre
dic
ted
th
e ex
iste
nce
of
an
tin
eutr
ino
s an
d s
ucc
essf
ull
y
des
crib
ed b
eta
dec
ay
s:
(,
)(
,1)
,
(,
)(
,1)
,
(8
86.7
15
+ −
−
→−
++
→+
++
→+
+=
sm
in).
�
e e
en
AZ
AZ
e
AZ
AZ
e
np
e
ν ννν ν ννν
ντ
ντ
ντ
ντ
It a
lso
pre
dic
ted
oth
er w
eak
pro
cess
es,
in p
art
icu
lar
the
inve
rse
bet
a d
ecay
s
such
as:
, .
+ −
+→
+
+→
+
e e
pe
n
ne
p
ν ννν ν ννν
Th
e d
irec
t d
etec
tio
n o
f (a
nti
)neu
trin
os
was
per
form
ed b
y C
. C
ow
an
an
d F
.
Rei
nes
in
1956
th
rou
gh
th
e co
inci
den
t o
bse
rva
tion
of
a p
osi
tro
n a
nd
a p
ho
ton
(in
fact
, w
ith
a f
ew µ µµµ
sec
del
ay)
emit
ted
in
su
bse
qu
ent
react
ion
s
112
113
.+
+→
+�
+→
+C
dC
de
pe
nn
νγ
νγ
νγ
νγ
5
Th
ey u
sed
th
e n
ucl
ear
react
or
(Sava
nn
ah
Riv
er, S
ou
th C
aro
lin
a)
as
an
an
tin
eutr
ino
so
urc
e a
nd
200 l
iter
s of
wate
r w
ith
40
kg o
f d
isso
lved
Cd
Cl 2
as
a
det
ecto
r.
F
red
eric
k R
ein
es w
as
aw
ard
ed t
he
199
5 N
ob
el P
rize
in
Ph
ysi
cs “
for
the
det
ecti
on
of
the
neu
trin
o”
.
T
he
det
ail
ed s
tud
ies
of
bet
a d
ecay
s es
tab
lish
ed t
he
−V
A s
tru
ctu
re o
f w
eak
inte
ract
ion
s le
ad
ing
to
pari
ty v
iola
tion
(C
. N
. Y
an
g a
nd
T.-
D. L
ee r
ecei
ved
th
e
1957
Nob
el P
rize
in
Ph
ysi
cs “
for
thei
r p
enet
rati
ng
in
ves
tiga
tio
n o
f th
e so
-ca
lled
pa
rity
law
s w
hic
h h
as
led
to i
mp
ort
an
t d
isco
ver
ies
rega
rdin
g t
he
elem
enta
ry
pa
rtic
les”
).
T
he
mod
ern
th
eory
of
wea
k i
nte
ract
ion
s is
a p
art
of
the
non
-ab
elia
n g
au
ge
theo
ry o
f el
ectr
ow
eak
in
tera
ctio
ns
dev
elo
ped
in
th
e 1
960’s
b
y S
hel
don
L.
Gla
sho
w, A
bd
us
Sa
lam
, S
tev
en W
ein
ber
g (
the
1979
Nob
el P
rize
in
Ph
ysi
cs “
for
thei
r co
ntr
ibu
tio
ns
to t
he
theo
ry o
f th
e u
nif
ied
wea
k a
nd
ele
ctro
mag
net
ic
inte
ract
ion
bet
wee
n e
lem
enta
ry p
art
icle
s, i
ncl
ud
ing,
inte
r a
lia
, th
e p
red
icti
on
of
the
wea
k n
eutr
al
curr
ent”
).
R
eno
rma
liza
bil
ity
of
the
sta
nd
ard
ele
ctro
wea
k m
od
el (
SM
) b
ase
d o
n t
he
gau
ge
gro
up
(2
)(1
)L
YS
UU
×(s
pon
tan
eou
sly b
rok
en t
o
(1) em
U)
wa
s p
roved
by
6
Ger
ard
us
‘t H
oo
ft a
nd
Mart
inu
s J. G
. V
eltm
an
in
th
e b
egin
nin
g o
f 1970
’s (
the
1999
Nob
el P
rize
in
Ph
ysi
cs “
for
elu
cid
ati
ng
th
e q
ua
ntu
m s
tru
ctu
re o
f
elec
tro
wea
k i
nte
ract
ion
s in
ph
ysi
cs”
).
Neu
trin
o f
lavo
rs
Up
to n
ow
th
ree
dis
tin
ct t
yp
es o
f n
eutr
inos
(an
d c
orr
esp
on
din
g a
nti
neu
trin
os)
hav
e b
een
det
ecte
d:
,,
eµ
τµ
τµ
τµ
τν
νν
νν
νν
νν
νν
ν (
,,
eµ
τµ
τµ
τµ
τν
νν
νν
νν
νν
νν
ν).
Ele
ctro
n n
eutr
ino
sa
nd
an
tin
eutr
inos
are
pro
du
ced
in
bet
a b
ecay
s (s
ee b
elo
w)
an
d i
n o
ther
dec
ay
s, e
. g.:
0
0
,,
,,
,.
ee
ee
ee
ee
Ke
Ke
ee
µµ
µµ
µµ
µµ
πν
πν
πν
πν
πν
πν
πν
πν
πν
πν
πν
πν
πν
πν
πν
πν
µν
νµ
νν
µν
νµ
νν
µν
νµ
νν
µν
νµ
νν
++
−−
++
−−
++
−−
→+
→+
→+
+→
++
→+
+→
++
It w
as
sup
pose
d t
ha
t ν
νν
νν
νν
ν≠
: 37
37
37
37
− −
+→
+
+→
+
(R.
Davis
, J
r., 195
2),
(R. D
avis
, Jr.
, D
. S
. H
arm
er,
an
d K
. C
. H
off
man
n, 19
68)
e e
Cl
Ar
e
Cl
Ar
e
ν ννν ν ννν
[th
e ex
per
imen
t w
as
pro
po
sed
by
B. P
on
teco
rvo
in
1946
].
7
Mu
on
(a
nti
)neu
trin
os:
,,
,,
,.
ee
KK
ee
µµ
µµ
µµ
µµ
µµ
µµ
µµ
µµ
µµ
µµ
µµ
µµ
πµ
νπ
µν
πµ
νπ
µν
πµ
νπ
µν
πµ
νπ
µν
µν
µν
µν
µν
µν
µν
µν
µν
µν
νµ
νν
µν
νµ
νν
µν
νµ
νν
µν
νµ
νν
++
−−
++
−−
++
−−
→+
→+
→+
→+
→+
+→
++
Th
e ex
per
imen
tal
evid
ence
th
at
eµ µµµ
νν
νν
νν
νν
≠ :
b
ut
NX
Ne
Xµ
µµ
µµ
µµ
µν
µν
νµ
νν
µν
νµ
ν−
−+
→+
+→
+
(L. L
eder
man
, M
. S
chw
arz
, a
nd
J.
Ste
inb
erg
er, 1962
).
Ta
u n
eutr
ino
s:
,
,
had
ron
s.
ee
τµ
ττ
µτ
τµ
ττ
µτ
τ τττ
τν
ντ
µν
ντ
νν
τµ
νν
τν
ντ
µν
ντ
νν
τµ
νν
τν
τν
τν
τν
−−
−−
−
→+
+→
++
→+
Th
e th
ird
ch
arg
ed l
epto
n τ τττ
− w
as
dis
cov
ered
in
19
75
at
Sta
nfo
rd. D
irec
t
evid
ence
fo
r th
e ta
u n
eutr
ino w
as
ob
tain
ed i
n 2
00
1 b
y t
he
DO
NU
T c
oll
ab
ora
tio
n
at
Fer
mil
ab
th
rou
gh
th
e ob
serv
ati
on
of
τ τττ-a
pp
eara
nce
in
nu
clea
r em
uls
ion
s:
.N
Xτ τττ
ντ
ντ
ντ
ντ
+→
+
8
Th
e ty
pe
of
a n
eutr
ino
ν ννν� i
s ca
lled
fla
vor.
In
th
e S
M t
her
e a
re t
hre
e n
eutr
ino
fla
vo
rs (
,,
eµ
τµ
τµ
τµ
τ=�
) ass
oci
ate
d w
ith
th
e co
rres
po
nd
ing c
ha
rged
lep
ton
s in
th
ree
elec
tro
wea
k-i
sosp
in d
ou
ble
ts:
,
,.
e
LL
Le
µ µµµτ τττ
ν νννν
νν
νν
νν
ν τ τττµ µµµ
−−
−
��
��
��
��
��
��
��
��
��
��
Neu
trin
os
ν ννν� h
ave
two t
yp
es o
f in
tera
ctio
ns
med
iate
d b
y c
ha
rged
W± b
oso
ns
(ch
arg
ed c
urr
ent
inte
ract
ion
s) a
nd
by
a n
eutr
al
0Z
boso
n (
neu
tra
l cu
rren
t
inte
ract
ion
s):
55
5
11
,2
22
1,
cos
2
CC
NC
W
gL
WW
gL
Z
µµ
µµ
µµ
µµ
µµ
µµ
µµ
µµ
µ µµµµ µµµ
γγ
γγ
γγ
γγ
γν
νγ
γν
νγ
γν
νγ
γν
νγ
γ γγγν
γν
νγ
νν
γν
νγ
νθ θθθ
−+
�
��
��
−−
=−
+
��
��
��
��
��
��
−=
−�
��
�
�
�
��
�
��
�
��
wh
ere
the
cou
pli
ng c
on
stan
t /s
inW
ge
θ θθθ=
, W
θ θθθ i
s th
e w
eak
(W
ein
ber
g)
an
gle
(2
sin
0.2
3)
Wθ θθθ�
, e
is
the
po
sitr
on
ele
ctri
c ch
arg
e.
B
y d
efin
itio
n, th
e fl
avo
r of
a n
eutr
ino i
s th
e ty
pe
of
the
cha
rged
lep
ton
th
at
is
con
nec
ted
to
th
e sa
me
cha
rged
cu
rren
t v
erte
x:
,
.W
Wν
νν
νν
νν
ν+
+−
−→
+→
+�
��
�
9
Nu
mb
er o
f li
gh
t n
eutr
inos
Th
e p
art
ial
wid
th o
f th
e d
eca
y Z
νν
νν
νν
νν
→+�
� is
ca
lcu
lab
le i
n t
he
SM
:
3
165
.91
22
MeV
FZ
Gm
ν νννπ πππ
Γ=
=.
Th
e in
vis
ible
wid
th o
f th
e d
eca
y i
nto
all
neu
trin
o f
ina
l st
ate
s:
499.0
1.5
.in
vM
eVN
νν
νν
νν
νν
Γ=
Γ=
±
It i
s o
bta
ined
ex
per
imen
tall
y f
rom
stu
die
s o
f si
ng
le-p
ho
ton
even
ts f
rom
th
e
react
ion
e
eν
νγ
νν
γν
νγ
νν
γ+
−→
or
by
su
btr
act
ing
th
e co
ntr
ibu
tion
of
all
vis
ible
ch
an
nel
s
(,
;,
,;
,,
,,
)Z
Zqq
eq
ud
sc
bµ
τµ
τµ
τµ
τ+
−→
→=
=��
� f
rom
th
e m
easu
red
to
tal
wid
th,
inv
vis
ZΓ
=Γ
−Γ
.
T
he
nu
mb
er o
f th
e li
gh
t act
ive
neu
trin
os
(th
at
hav
e th
e u
sua
l el
ectr
ow
eak
inte
ract
ion
s) i
s (s
ee F
ig. 1):
3
.01.
inv
Nν ννν
ν ννν
Γ=
=Γ
PD
G (
2004
):
2.9
94
0.0
12
(),
2.9
20
.07
().
inv
SM
fit
s to
LE
P d
ata
Dir
ect
mea
sure
men
t of
N N
ν ννν ν ννν
=±
=±
Γ
1
0
F
ig.
1.
1
1
Ch
irali
ty a
nd
hel
icit
y
In 1
950
’s i
t w
as
dis
cov
ered
th
at
all
neu
trin
os
hav
e (w
ith
in e
xp
erim
enta
l
un
cert
ain
ties
) sp
in a
nti
pa
rall
el t
o t
hei
r m
om
entu
m, w
hil
e fo
r a
ll a
nti
neu
trin
os
spin
an
d m
om
entu
m a
re p
ara
llel
. T
his
is
a c
on
seq
uen
ce o
f th
e V
A−
str
uct
ure
of
the
wea
k c
urr
ents
:
(
),2
,,
CC
LL
LL
gL
Wj
Wj
jj
µµ
µµ
µµ
µµ
µµ
µµ
µµ
µµ
µµ
µµ
µµ
µµ
µµ
µµ
µµ
µµ
γν
νγ
γν
νγ
γν
νγ
γν
νγ
−−
++
−+
=−
+
==
��
��
��
��
wh
ere
( ((() )))
( ((() )))
55
50
25
01
23
55
55
5
11
1,
,;
22
2
,,
,.
LL
L
iI
µµ
µµ
µµ
µµ
γγ
γγ
γγ
γγ
γγ
γγ
ψψ
ψψ
ψψ
ψψ
γψ
ψψ
ψψ
ψψ
ψγ
ψψ
ψψ
ψψ
ψψ
γψ
ψψ
ψψ
ψψ
ψγ
γγ
γγ
γγ
γγ
γγ
γγ
γγ
γγ
γγ
γγ
γγ
γγ
γγ
γγ
γγ
γγ
γγ
γγ
γγ
γγ
γγ
γγ
γγ
γγ
+ +++
+ +++
−−
+−
−+
−−
+−
−+
==
==
==
==
==
==
==
==
==
==
==
−=
==
=−
==
==
−=
==
=−
==
Ch
ira
lity
is
eigen
va
lue
of
th
e o
per
ato
r 5
γ γγγ:
1
2
55
5,;
1,
;2
,
.
LL
RR
RL
R
LL
RR
RL
LR
µµ
µµ
µµ
µµ
µµ
µµ
µµ
µµ
µµ
µµ
µµ
µµ
γψ
ψγ
ψψ
γψ
ψγ
ψψ
γψ
ψγ
ψψ
γψ
ψγ
ψψ
γ γγγψ
ψψ
ψψ
ψψ
ψψ
ψψ
ψψ
ψψ
ψψ
ψψ
ψ
ψγ
ψψ
γψ
ψγ
ψψ
γψ
ψγ
ψψ
γψ
ψγ
ψψ
γψ
ψγ
ψψ
γψ
ψγ
ψψ
γψ
ψψ
ψψ
ψψ
ψψ
ψψ
ψψ
ψψ
ψψ
ψψ
ψψ
ψψ
ψψ
=−
=+
=−
=+
=−
=+
=−
=+
+ +++=
=+
==
+=
=+
==
+
∂=
∂+
∂∂
=∂
+∂
∂=
∂+
∂∂
=∂
+∂
=+
=+
=+
=+
Lψ ψψψ
an
d
Rψ ψψψ
are
ca
lled
lef
t-h
an
ded
an
d r
igh
t-h
an
ded
fie
lds
resp
ecti
vel
y.
T
he
SM
is
a c
hir
al
ga
ug
e th
eory
, si
nce
th
ere
are
L-d
ou
ble
ts a
nd
R-s
ing
lets
,
,
;,
,et
c.e
LL
RR
R
LL
ue
ud
deν ννν �
��
��
��
��
��
��
��
��
��
��
��
��
��
��
��
��
��
��
��
��
��
��
��
�,
wh
ich
hav
e d
iffe
ren
t w
eak
in
tera
ctio
ns.
T
he
Dir
ac
equ
ati
on
for
a m
ass
ive
pa
rtic
le w
ith
4-m
om
entu
m
(,p
)p
Eµ µµµ
= ===,
( (((
) )))0
pm
µ µµµµ µµµ
γψ
γψ
γψ
γψ
−=
−=
−=
−=
,
can
be
dec
om
po
sed
as
foll
ow
s:
1
3
( ((() )))
0 0
1/2
22
ˆ,
ˆ,
,|p
|,
LL
R
RR
L
Em
hp
p
Em
hp
p
Em
pp
ψψ
γψ
ψψ
γψ
ψψ
γψ
ψψ
γψ
ψψ
γψ
ψψ
γψ
ψψ
γψ
ψψ
γψ
=−
+=
−+
=−
+=
−+
=+
−=
+−
=+
−=
+−
=+
==
+=
=+
==
+=
wh
ere
the
hel
icit
y o
per
ato
r is
5
00
pˆ
,.
0p
kk
k
k
hσ σσσ
γγ
γγ
γγ
γγ
γγ
γγ
σ σσσ
��
��
��
��
Σ⋅
Σ⋅
Σ⋅
Σ⋅
=Σ
==
=Σ
==
=Σ
==
=Σ
==�
��
��
��
��
��
��
��
�
Hel
icit
y is
a c
on
serv
ed q
uan
tum
nu
mb
er, in
con
tra
st w
ith
ch
ira
lity
th
at
is
con
serv
ed o
nly
in
th
e m
ass
less
lim
it. F
or
ma
ssle
ss p
art
icle
s h
elic
ity
an
d
chir
ali
ty a
re i
den
tica
l:
ˆ
ˆ,
.L
LR
Rh
hψ
ψψ
ψψ
ψψ
ψψ
ψψ
ψψ
ψψ
ψ=
−=
=−
==
−=
=−
=
1
4
Fo
r a
ma
ssiv
e p
art
icle
, ch
irali
ty s
tate
s are
mix
ture
s o
f h
elic
ity s
tate
s, a
nd
in
th
e
ult
rare
lati
vis
tic
lim
it (
/1
�m
E)
,
;2
2
ˆ.
��
LR
mm
EE
hψψ
ψψ
ψψ
ψψ
ψψ
ψψ
ψψ
ψψ
ψψ
ψψ
ψψ
ψψ
ψψ
ψψ
ψψ
ψψ
−+
+−
−+
+−
−+
+−
−+
+−
±±
±±
±±
±±
++
++
++
++
=±
=±
=±
=±
1
5
( ((() )))
( ((() )))
( ((() )))
()
1/2
55
(2)
(,
)
p,
,,
1.
p
11
11
11
11
,.
11
11
22
22
1,
2 1.
2
For
1:
2
11
22
ipx
p LR
LL
RR
L
EV
up
he
Em
wu
ph
wh
wh
Em
hw
PP
wu
Pu
Em
hE
mw
wu
Pu
Em
hE
mw
m E
wh
hu
Ew
ψ ψψψ
σ σσσ
γγ
γγ
γγ
γγ
δ δδδ
δ δδδ
+−
−⋅
+−
−⋅
+−
−⋅
+−
−⋅
= ===
��
��
��
��
+ +++⋅ ⋅⋅⋅
==
=±
==
=±
==
=±
==
=±
��
��
��
��
��
��
��
��
− −−−�
��
��
��
�
− −−−�
��
��
��
��
��
��
��
�−
+−
+−
+−
+=
==
==
==
==
==
==
==
=�
��
��
��
��
��
��
��
�− −−−�
��
��
��
��
��
��
��
�
��
��
��
��
==
+−
−=
=+
−−
==
+−
−=
=+
−−
��
��
��
��
− −−−�
��
��
��
�
��
��
��
��
==
++
−=
=+
+−
==
++
−=
=+
+−
��
��
��
��
��
��
��
��
= ===
� ���−
+−
+−
+−
+�
��
��
��
�+ +++
��
��
��
�� � ���
− −−−�
��
��
��
� � ���
� ���
� ���,
11
.2
2R
wh
hu
Ew
δ δδδ
� ��� � ��� � ���
��
��
��
��
+−
+−
+−
+−
��
��
��
��
+ +++�
��
��
��
���
��
��
��
��
��
��
�� �
��
��
��
�� ���
1
6
T
he
dif
fere
nce
bet
wee
n c
hir
ali
ty a
nd
hel
icit
y h
as
imp
ort
an
t co
nse
qu
ence
s.
Th
e im
pre
ssiv
e m
an
ifes
tati
on
of
the
VA
− −−− s
tru
ctu
re o
f th
e ch
arg
ed w
eak
curr
ent
is t
he
hel
icit
y s
up
pre
ssio
n i
n π πππ
dec
ay
s:
( ((() )))
22
22
4
22
4
exp
()
1.2
810
,(
)
1.2
30
0.0
04
10
.
ee
ee
mm
mR
mm
m
R
π πππ
µ µµµµ
πµ
µπ
µµ
πµ
µπ
µ
πν
πν
πν
πν
πµ
νπ
µν
πµ
νπ
µν
++
++
++
++
− −−−
++
++
++
++
− −−−
��
��
��
��
��
��
��
��
Γ→
+−
Γ→
+−
Γ→
+−
Γ→
+−
==
=×
==
=×
==
=×
==
=×
��
��
��
��
��
��
��
��
��
��
��
���
��
��
��
�Γ
→+
−Γ
→+
−Γ
→+
−Γ
→+
−�
��
��
��
��
��
��
��
�
=±
×=
±×
=±
×=
±×
Th
e d
ecay
am
pli
tud
e is
pro
po
rtio
na
l to
th
e ad
mix
ture
of
neg
ati
ve
hel
icit
y i
n t
he
rig
ht-
ha
nd
ed s
tate
of
a c
ha
rged
lep
ton
(se
e F
ig. 2
):
(
)�
��
�A
mπ
νπ
νπ
νπ
ν+
++
++
++
+→
+→
+→
+→
+.
F
ig. 2. T
he
dec
ay o
f a
ch
arg
e p
ion
in
its
res
t fr
am
e.
T
he
chir
al
stru
ctu
re o
f w
eak
in
tera
ctio
ns
op
ens
a p
oss
ibil
ity
th
at
the
sta
tes
ν ννν
an
d ν ννν
are
th
e tw
o d
iffe
ren
t h
elic
ity
sta
tes
of
a M
ajo
ran
a n
eutr
ino
Mν ννν
th
at
is
iden
tica
l to
its
an
tip
art
icle
:
1
7
(
1),
(1).
MM
hh
νν
νν
νν
νν
νν
νν
νν
νν
==
−=
=+
==
−=
=+
==
−=
=+
==
−=
=+
In e
xp
erim
ents
wit
h u
ltra
rela
tiv
isti
c (a
nti
)neu
trin
os,
it
is p
ract
ica
lly i
mp
oss
ible
to d
isti
ng
uis
h b
etw
een
Dir
ac
()
DD
νν
νν
νν
νν
≠ ≠≠≠ a
nd
Majo
ran
a (
)M
Mν
νν
νν
νν
ν≡ ≡≡≡
neu
trin
os
du
e
to s
tro
ng
(2
(/
)m
E�
) su
pp
ress
ion
of
“w
ron
g h
elic
ity
” s
tate
s.
Ob
serv
ati
on
of
neu
trin
o o
scil
lati
on
s:
no
nze
ro n
eutr
ino
ma
sses
an
d m
ixin
g
T
he
rece
nt
(1998
-200
2)
ob
serv
ati
on
of
neu
trin
o o
scil
lati
on
s (p
red
icte
d b
y B
.
Po
nte
corv
o i
n 1
957)
imp
lies
th
at
neu
trin
os
are
ma
ssiv
e an
d m
ixed
pa
rtic
les.
Th
e n
eutr
ino ν ννν�
of
flav
or
,
,e
µτ
µτ
µτ
µτ
= ===�
is
th
e su
per
po
siti
on
of
neu
trin
os
i
ν ννν w
ith
def
init
e m
ass
es
im
,
,i
i
i
Uν
νν
νν
νν
ν= ===� ���
��
wh
ere
i
U�
‘s
form
th
e n
eutr
ino
mix
ing
ma
trix
. T
he
flavo
r n
eutr
ino
ν ννν�
is
crea
ted
in
ass
oci
ati
on
wit
h t
he
charg
ed l
epto
n
+ +++�
i
n t
he
dec
ay
W
ν ννν+
++
++
++
+→
+→
+→
+→
+�
�.
1
8
Th
e o
scil
lati
on
pro
ba
bil
ity (
see
Fig
. 3)
22
*2
22
**
2
()
()
exp
2
2R
eex
p,
2
jj
j
j
jj
jj
kk
jk
jj
k
LP
LU
Uim
E
LU
UU
UU
Ui
mE
αβ
βα
βα
αβ
βα
βα
αβ
βα
βα
αβ
βα
βα
αβ
αβ
αβ
αβ
αβ
αβ
αβ
αβ
αβ
αβ
αβ
αβ
νν
νν
νν
νν
νν
νν
νν
νν
> >>>
��
��
��
��
→=
=−
→=
=−
→=
=−
→=
=−�
��
��
��
��
��
��
��
�
��
��
��
��
=+
−∆
=+
−∆
=+
−∆
=+
−∆
��
��
��
��
��
��
��
��
� ���
��
��
��
��
wh
ere
2
22,
jkj
km
mm
∆=
−∆
=−
∆=
−∆
=−
E
is t
he
neu
trin
o e
ner
gy,
L
is t
he
dis
tan
ce b
etw
een
a
sou
rce
an
d a
det
ecto
r. T
he
exp
an
sion
of
the
neu
trin
o m
om
entu
m
22
1/2
()
jj
pE
m=
−=
−=
−=
− i
n
2(
/)
jm
E h
as
bee
n u
sed
:
2
exp
()
exp
2
jiE
L
j
mip
Le
iL
E
��
��
��
��
− −−−�
��
��
��
��
��
��
��
��
��
��
��
��
.
1
9
Fig
.3
. N
eu
trin
o f
lav
or
cha
ng
e i
n v
acu
um
.
Am
pW
W
So
urc
eT
arget
να
νβ
lβ-
lα+
= Σ
Am
pi
WW
So
urc
eT
arget
νi
lβ-
lα+
Uβ
iU
αi*
exp[-
imi
]
L 2E
2
2
0
I
n t
he
sim
ple
st c
ase
of
two
-neu
trin
o m
ixin
g
cos
sin
sin
cos
Uθ
θθ
θθ
θθ
θ
θθ
θθ
θθ
θθ
��
��
��
��
= ===�
��
��
��
�− −−−�
��
��
��
�,
the
osc
illa
tio
n p
rob
ab
ilit
ies
(see
Fig
. 4
)
2
22
221
osc
1(
)si
n2
sin
sin
21
cos
2;
42
()
1(
),mL
PL
EL
PP
αβ
αβ
αβ
αβ
αα
αβ
αα
αβ
αα
αβ
αα
αβ
νν
θθ
πν
νθ
θπ
νν
θθ
πν
νθ
θπ
νν
νν
νν
νν
νν
νν
νν
νν
�
�
�
�
��
��
��
��
��
��
��
��
∆ ∆∆∆→
==
−→
==
−→
==
−→
==
−
�
�
�
��
��
��
��
��
��
��
��
��
��
��
��
��
��
��
��
��
�
�
�
→=
−→
→=
−→
→=
−→
→=
−→
wh
ere
the
osc
illa
tio
n l
eng
th
osc
22
2
21
21
4(M
eV)
2.4
8m
.(e
V)
EE
Lm
m
π πππ=
==
==
==
=∆
∆∆
∆∆
∆∆
∆
2
1
Fig
. 4
.
2
2
U
p t
o n
ow
th
e o
scil
lati
on
s h
av
e b
een
ob
serv
ed f
or
sola
r (H
om
esta
ke,
SA
GE
, G
AL
LE
X-G
NO
, S
NO
:
()
eµ
τµ
τµ
τµ
τν
νν
νν
νν
νν
νν
ν→ →→→
),
atm
osp
her
ic (
Su
per
-Kam
iok
an
de:
µ
τµ
τµ
τµ
τν
νν
νν
νν
ν→ →→→
) ,
rea
cto
r (K
am
LA
ND
:
eµ µµµ
νν
νν
νν
νν
→ →→→)
[so
l⇔ ⇔⇔⇔
],
an
d a
ccel
era
tor
(K2K
:
µτ
µτ
µτ
µτ
νν
νν
νν
νν
→ →→→)[
atm
⇔ ⇔⇔⇔]
neu
trin
os
[fo
r a
rev
iew
, se
e P
DG
-200
4].
T
he
sin
uso
ida
l L
/E d
epen
den
ce o
f th
e su
rviv
al
pro
bab
ilit
y
22
22
(km
)(
)1
sin
sin
1.2
7(e
V)
(GeV
)e
e
LP
mE
νν
θν
νθ
νν
θν
νθ
��
��
��
��
→=
−∆
→=
−∆
→=
−∆
→=
−∆
��
��
��
��
��
��
��
��
ha
s b
een
rec
entl
y c
on
firm
ed b
y K
am
LA
ND
(h
ep-e
x/0
40
6035,
see
Fig
. 5).
2
3
20
30
40
50
60
70
80
0
0.2
0.4
0.6
0.81
1.2
1.4
(k
m/M
eV)
eν
/E0
L
Ratio2
.6 M
eV p
rom
pt
anal
ysi
s th
resh
old
Kam
LA
ND
dat
a
bes
t-fi
t o
scil
lati
on
bes
t-fi
t d
ecay
bes
t-fi
t d
eco
her
ence
Fig
. 5
. R
ati
o o
f th
e ob
serv
ed a
nti
neu
trin
o s
pec
tru
m t
o t
he
exp
ecta
tion
for
no-o
scil
lati
on
ver
sus
L/E
. T
he
curv
es s
how
th
e ex
pec
tati
on
for
the
bes
t-fi
t osc
illa
tion
, b
est-
fit
dec
ay
an
d b
est-
fit
dec
oh
eren
ce m
od
els
tak
ing i
nto
acc
ou
nt
the
ind
ivid
ual
tim
e-d
epen
den
t fl
ux
vari
ati
on
s of
all
rea
ctors
an
d d
etec
tor
effe
cts.
Th
e d
ata
poin
ts a
nd
mod
els
are
plo
tted
2
4
wit
h L
=180 k
m, as
if a
ll a
nti
neu
trin
os
det
ecte
d i
n K
am
LA
ND
wer
e d
ue
to a
sin
gle
react
or
at
this
dis
tan
ce.
Th
e m
inim
al
sch
eme
of
thre
e n
eutr
ino
mix
ing,
3
12
3
1
;j
j
j
Um
mm
νν
νν
νν
νν
= ===
=<
<=
<<
=<
<=
<<
� ����
�,
pro
vid
es t
wo
in
dep
end
ent
2
m∆ ∆∆∆
an
d a
llow
to d
escr
ibe
sola
r an
d a
tmo
sph
eric
neu
trin
o o
scil
lati
on
da
ta. T
her
e are
tw
o p
oss
ibil
itie
s fo
r h
iera
rch
y o
f n
eutr
ino
ma
ss-s
qu
are
d d
iffe
ren
ces:
H1
: 2
22
2
21
sol
32
atm
;m
mm
m∆
∆∆
∆∆
∆∆
∆∆
∆∆
∆∆
∆∆
∆�
��
H2
: 2
22
2
32
sol
21
atm
mm
mm
∆∆
∆∆
∆∆
∆∆
∆∆
∆∆
∆∆
∆∆
��
�.
F
or
H1
, th
e sh
ort
osc
illa
tio
ns
are
op
era
tin
g b
ut
the
lon
g o
nes
hav
e n
ot
dev
elop
ed:
2
2
32
21
/1,
/1
mL
Em
LE
∆>
∆∆
>∆
∆>
∆∆
>∆
��
,
2
5
( ((() )))
( ((() )))
( ((() )))
22
22
33
32
22
22
33
32
2
3
4si
n,
4
14
1si
n,
4
1.
LP
UU
mE
LP
UU
mE
UU
IU
αβ
αβ
αβ
αβ
αβ
αβ
αβ
αβ
αα
αα
αα
αα
αα
αα
αα
αα
α αααα ααα
νν
νν
νν
νν
νν
νν
νν
νν
+ +++
��
��
��
��
→∆
→∆
→∆
→∆�
��
��
��
��
��
��
��
�
��
��
��
��
→−
−∆
→−
−∆
→−
−∆
→−
−∆�
��
��
��
��
��
��
��
�
=→
==
→=
=→
==
→=
� ���� �
(Fo
r H
2:
22
22
32
21
31
,.
mm
UU
αα
αα
αα
αα
∆→
∆→
∆→
∆→
∆→
∆→
∆→
∆→
)
Fo
r v
ery
lon
g b
ase
lin
e ex
per
imen
ts,
2 32
1,
2mL
E
∆ ∆∆∆�
the
av
erag
ed (
ov
er f
ast
osc
illa
tio
ns)
su
rviv
al
pro
ba
bil
ity
( (((
) )))( (((
) )))24
22
22
2
33
12
21
14
sin
2e
ee
ee
e
LP
UU
UU
mE
νν
νν
νν
νν
�
�
�
�
��
��
��
��
→=
+−
−∆
→=
+−
−∆
→=
+−
−∆
→=
+−
−∆�
��
��
��
�
�
�
�
��
��
��
��
��
�
�
�
.
Th
e st
an
dard
pa
ram
eter
izati
on
of
the
Pon
teco
rvo
–M
ak
i–N
ak
ag
aw
a–
Sa
kata
mix
ing m
atr
ix:
2
6
1
2
12
3
12
3
12
3
13
13
12
12
23
23
12
12
23
23
13
13
10
00
01
00
00
10
00
0,
00
00
10
0
ee
e
i
i
ii
UU
U
UU
UU
UU
U
cs
ec
s
cs
sc
e
sc
se
ce
µµ
µµ
µµ
µµ
µµ
µµ
ττ
ττ
ττ
ττ
ττ
ττ
δ δδδ
α ααα
α αααδ δδδ
��
��
��
��
��
��
��
��
= ===�
��
��
��
��
��
��
��
��
��
��
��
�
��
��
��
��
��
��
��
��
���
��
���
���
��
���
��
��
��
��
��
��
��
��
���
��
���
���
��
���
=−
=−
=−
=−
��
��
��
��
��
��
��
��
���
��
���
���
��
���
���
��
���
���
��
���
��
��
��
�� �
��
��
��
�−
−−
−−
−−
−�
���
���
��
���
���
��
��
��
��
� ��
��
��
��
wh
ere
12
13
23
sin
,co
s;
,,
jkjk
jkjk
sc
θθ
θθ
θθ
θθ
θθ
θθ
θθ
θθ
θθ
θθ
==
==
==
==
are
th
e m
ixin
g a
ng
les;
δ δδδ i
s th
e C
P
vio
lati
ng
Dir
ac
ph
ase
; 1
2,
αα
αα
αα
αα
are
Ma
jora
na p
hase
s.
T
he
Ma
jora
na p
ha
ses
are
com
mon
to
an
en
tire
co
lum
n o
f th
e m
ixin
g m
atr
ix,
an
d t
her
efo
re t
hey
ha
ve
no
eff
ect
on
neu
trin
o o
scil
lati
on
s.
In t
his
pa
ram
etri
zati
on
(om
itti
ng
th
e M
ajo
ran
a p
hase
s),
1
12
13
212
13
313
313
23
313
23
cos
cos
,si
nco
s,
sin
;
cos
sin
,co
sco
s.
i
ee
eU
UU
e
UU
δ δδδ
µτ
µτ
µτ
µτ
θθ
θθ
θθ
θθ
θθ
θθ
θθ
θθ
θθ
θθ
θθ
θθ
θθ
θθ
θθ
θθ
θθ
θθ
− −−−=
==
==
==
==
==
=
==
==
==
==
Fro
m e
xp
erim
enta
l d
ata
,
2
22
31
3si
n5
10
(0
)e
Uθ θθθ
− −−−=
<×
=<
×=
<×
=<
�
.
2
7
In t
he
lead
ing a
pp
rox
imati
on
, n
eutr
ino o
scil
lati
on
s in
atm
osp
her
ic a
nd
so
lar
ran
ges
of
2m
∆ ∆∆∆a
re d
escr
ibed
by
tw
o-n
eutr
ino
form
ula
s w
ith
2
22
22
22
2
atm
32
atm
23
sol
21
sol
12
,si
nsi
n a
nd
,si
nsi
n,
mm
mm
θθ
θθ
θθ
θθ
θθ
θθ
θθ
θθ
∆∆
∆∆
∆∆
∆∆
∆∆
∆∆
∆∆
∆∆
��
��
resp
ecti
vel
y.
At
99.7
3%
C.L
. (3�)
th
e e
ffe
cti
ve t
wo
-neu
trin
o m
ixin
g p
ara
mete
rs a
re
co
nstr
ain
ed
in t
he
ran
ges:
32
23
22
atm
atm
52
25
22
sol
sol
1.4
10
eV5.1
10
eV,
sin
20.8
6;
5.4
10
eV5.4
10
eV,
0.3
0ta
n0.6
4;
m m
θ θθθ
θ θθθ
−−
−−
−−
−−
−−
−−
−−
−−
×<
∆<
×>
×<
∆<
×>
×<
∆<
×>
×<
∆<
×>
×<
∆<
×<
<×
<∆
<×
<<
×<
∆<
×<
<×
<∆
<×
<<
the
allo
wed
ra
ng
es f
or
the e
lem
en
ts o
f th
e m
ixin
g m
atr
ix:
0.7
60.8
80.4
70.6
20
.00
0.2
2
=0.0
90.6
20.2
90.7
90
.55
0.8
5
0.1
10.6
20.3
20.8
00.5
10.8
3
U
÷÷
÷÷
÷÷
÷÷
÷÷
÷÷
��
��
��
��
��
��
��
��
÷÷
÷÷
÷÷
÷÷
÷÷
÷÷
��
��
��
��
��
��
��
��
÷÷
÷÷
÷÷
÷÷
÷÷
÷÷
��
��
��
��.
Th
e b
est
fit:
2
8
2-3
22
atm
atm
25
22
sol
sol
bf
2.6
10
eV,
sin
21;
6.9
10
eV,
tan
0.4
3;
0.8
40.5
50.0
0
0.3
9
0.5
90.7
1.
0.3
90.5
90.7
1
m
m
U
θ θθθ
θ θθθ− −−−
∆=
×=
∆=
×=
∆=
×=
∆=
×=
∆=
×=
∆=
×=
∆=
×=
∆=
×=
��
��
��
��
��
��
��
��
=−
=−
=−
=−�
��
��
��
��
��
��
��
�− −−−
��
��
��
��
Th
e n
eu
trin
o m
ixin
g m
atr
ix (
wit
h a
ll e
lem
en
ts larg
e e
xc
ep
t 3
eU�
is v
ery
dif
fere
nt
fro
m t
he
qu
ark
mix
ing
matr
ix, in
wh
ich
mix
ing
is v
ery
s
ma
ll.
Lim
its
on
th
e n
eutr
ino
ma
sses
�
Fro
m t
riti
um
bet
a d
ecay
33
eH
He
eν ννν
− −−−→
++
→+
+→
++
→+
+ (
fitt
ing
th
e sh
ap
e o
f th
e b
eta
spec
tru
m):
1/2
22
2.0
5(2
.2)
eV [
Tro
itsk
(M
ain
z),
95%
CL
].ek
k
k
mU
mβ βββ
��
��
��
��
=<
=<
=<
=<
��
��
��
��
��
��
��
��
� ���
2
9
It i
mp
lies
an
up
per
lim
it o
n t
he
min
ima
l n
eutr
ino
ma
ss (
21
ek
k
U= ===
� ���):
1
.m
mβ βββ
≤ ≤≤≤
� F
rom
th
e o
scil
lati
on
s o
f a
tmosp
her
ic n
eutr
ino
s (a
ssu
min
g
12
30
mm
m≤
<<
≤<
<≤
<<
≤<
<):
( (((
) )))1/2
22
3atm
510
eV
.m
m− −−−
>∆
×>
∆×
>∆
×>
∆×
�
� F
rom
rec
ent
cosm
olo
gic
al
da
ta (
the
hig
h-p
reci
sio
n C
MB
R d
ata
of
the
WM
AP
sate
llit
e co
mb
ined
wit
h o
ther
ast
ron
om
ical
da
ta a
nd
som
e co
smo
log
ica
l
ass
um
pti
on
s):
0.7
1 e
V (
at
95%
CL
),k
k
m< <<<
� ���
wh
ere
th
e s
um
ru
ns o
ve
r all t
he
lig
ht
(1 M
eVk
m< <<<�
) n
eu
trin
o m
as
s
eig
en
sta
tes
th
at
were
in
th
erm
al eq
uilib
riu
m in
th
e e
arl
y U
niv
ers
e. F
or
jus
t th
ree lig
ht
kν ννν
, ta
kin
g in
to a
cco
un
t 2
22
sol
atm
1eV
mm
∆∆
∆∆
∆∆
∆∆
��
, it
im
plie
s
0.2
3eV
.k
m< <<<
Th
en f
or
the
hea
vie
st
3ν ννν
,
3
0
3
0.0
5 e
V0.2
3 e
V.
m<
<<
<<
<<
<
� F
rom
neu
trin
ole
ss d
ou
ble
bet
a d
eca
y (
,)
(,
2)
AZ
AZ
ee
−−
−−
−−
−−
→+
++
→+
++
→+
++
→+
++
, th
e
effe
ctiv
e M
ajo
ran
a m
ass
2
0.3
1.3
eV
eeek
k
k
mU
m=
<÷
=<
÷=
<÷
=<
÷� ���
(H
eid
elb
erg–M
osc
ow
, IG
EX
).
[Th
e larg
e u
nce
rta
inty
is
du
e t
o t
he
dif
fic
ult
y o
f c
alc
ula
tin
g t
he n
uc
lea
r m
atr
ix e
lem
en
t o
f th
e d
ecay].
Th
e n
atu
re o
f n
eutr
ino
ma
sses
: D
irac
or
Ma
jora
na?
T
o b
e D
ira
c or
Ma
jora
na? T
ha
t is
on
e o
f th
e m
ain
un
solv
ed q
ues
tio
ns
of
pa
rtic
le p
hysi
cs.
Th
e n
eutr
ino o
scil
lati
on
s d
o n
ot
pro
be
the
natu
re o
f th
e m
ass
.
3
1
Th
e D
irac
neu
trin
o c
arr
ies
the
lep
ton
nu
mb
er t
ha
t d
isti
ng
uis
hes
it
fro
m t
he
an
tin
eutr
ino
, an
d t
he
Dir
ac
neu
trin
o m
ass
is
gen
era
ted
ju
st l
ike
the
qu
ark
an
d
cha
rged
lep
ton
mass
es v
ia t
he
sta
nd
ard
Hig
gs
mec
ha
nis
m.
Th
e D
irac
mass
ter
m
5
,
();
1(1
).
2
D
DD
RL
LR
LR
Lm
mν
νν
νν
νν
νν
νν
νν
νν
νν
νν
νν
νν
ν
νγ
νν
γν
νγ
νν
γν
=−
=−
+=
−=
−+
=−
=−
+=
−=
−+
= ===�
Her
e v/
2,
Dm
y= ===
y i
s th
e Y
ukaw
a c
ou
plin
g a
nd
� is
th
e v
acu
um
exp
ec
tati
on
va
lue o
f th
e H
igg
s f
ield
. T
he
sp
ino
r L
Rν
νν
νν
νν
νν
νν
ν=
+=
+=
+=
+ h
as
4
ind
ep
en
den
t co
mp
on
en
ts.
Th
e M
ajo
ran
a m
ass
ter
ms
1
1(
),(
).2
2
Mc
cM
cc
LL
LL
LL
RR
RR
RR
Lm
Lm
νν
νν
νν
νν
νν
νν
νν
νν
νν
νν
νν
νν
νν
νν
νν
νν
=−
+=
−+
=−
+=
−+
=−
+=
−+
=−
+=
−+
Her
e th
e ch
arg
ed c
on
jug
ate
d s
pin
or
is d
efin
ed a
s fo
llow
s:
3
2
02
0
11
51
5
,,
,,
.
cT
T
TT
T
CC
Ci
CC
CC
CC
CC
µµ
µµ
µµ
µµ
ψψ
γψ
γγ
ψψ
γψ
γγ
ψψ
γψ
γγ
ψψ
γψ
γγ
γγ
γγ
γγ
γγ
γγ
γγ
γγ
γγ
∗ ∗∗∗
−+
−−
+−
−+
−−
+−
− −−−
==
==
==
==
==
==
=−
==
−=
−=
=−
=−
==
−=
−=
=−
= ===
Sin
ce
51
55
5,
0,
TC
Cµ
µµ
µµ
µµ
µγ
γγ
γγ
γγ
γγ
γγ
γγ
γγ
γγ
γγ
γγ
γγ
γ− −−−
=+
==
+=
=+
==
+=
( ((() )))
( ((() )))
( ((() )))
( ((() )))
( ((() )))
( ((() )))
5 5
11
,2 1
1.
2
cc
cc
LL
R
cc
cc
RR
L
ψψ
γψ
ψψ
ψγ
ψψ
ψψ
γψ
ψψ
ψγ
ψψ
ψψ
γψ
ψψ
ψγ
ψψ
ψψ
γψ
ψψ
ψγ
ψψ
≡=
+=
≡=
+=
≡=
+=
≡=
+=
≡=
−=
≡=
−=
≡=
−=
≡=
−=
Th
e M
ajo
ran
a m
ass
ter
m v
iola
tes
lep
ton
nu
mb
er b
y t
wo
un
its,
2.
L∆
=±
∆=
±∆
=±
∆=
±
Th
e M
ajo
ran
a n
eutr
ino
is
a t
rue
neu
tral
part
icle
id
enti
cal
to i
ts
an
tip
art
icle
:
(
E. M
ajo
ran
a,
1937).
cν
νν
νν
νν
ν= ===
Fo
r th
e M
ajo
ran
a f
ield
,
3
3
c
cc
c
LR
LR
LL
ψψ
ψψ
ψψ
ψψ
ψψ
ψψ
ψψ
ψψ
ψψ
ψψ
ψψ
ψψ
ψψ
ψψ
ψψ
ψψ
=+
=+
=+
==
+=
+=
+=
=+
=+
=+
==
+=
+=
+=
,
sin
ce
( ((() )))
.c
c
LR
Rψ
ψψ
ψψ
ψψ
ψψ
ψψ
ψ=
==
==
==
= T
his
fie
ld d
ep
en
ds o
nly
on
th
e t
wo
in
dep
en
den
t
co
mp
on
en
ts o
f L
ψ ψψψ.
T
he
tota
l D
ira
c-M
ajo
ran
a m
ass
ter
m
1
()
H.c
.,2
LD
LD
MD
MM
c
LR
LR
c
DR
R
mm
LL
LL
mm
ν νννν
νν
νν
νν
νν ννν
+ +++�
���
���
��
���
���
�=
++
=−
+=
++
=−
+=
++
=−
+=
++
=−
+�
���
���
��
���
���
��
���
���
��
���
���
�
wh
ere
Lν ννν
an
d
Rν ννν
are
in
dep
end
ent.
Th
e D
-M m
ass
ter
m i
n t
he
matr
ix f
orm
1H
.c.,
2
,.
DM
c LL
LL
D
Lc R
DR
LN
MN
mm
NM
mm
ν ννν ν ννν
+ +++=
−+
=−
+=
−+
=−
+
��
��
��
��
��
��
��
��
==
==
==
==
��
��
��
��
��
��
��
��
��
��
��
��
��
��
��
��
3
4
Dia
gon
ali
zati
on
giv
es t
he
ma
ss e
igen
sta
tes:
11
22
0,
,,
0
LT
L
L
mU
MU
UU
IN
Um
ν ννν ν ννν+ +++
��
��
��
��
��
��
��
��
==
==
==
==
==
==
��
��
��
��
��
��
��
��
��
��
��
��
��
��
��
��
wh
ere
0k
m≥ ≥≥≥
.
F
or
the
sim
ple
st c
ase
of
a r
eal
ma
ss m
atr
ix M
:
( ((() )))
21
2
21
11
22
22
22
1,2
0co
ssi
n,
1;0
sin
cos
00
,0
0
21
tan
2,
4.
2
k
T
D
LR
LR
D
RL
UO
mm
UM
Um
m
mm
mm
mm
mm
m
ρ ρρρθ
θθ
θθ
θθ
θρ
ρρ
ρρ
ρρ
ρρ ρρρ
θθ
θθ
θθ
θθ ρ ρρρ
ρ ρρρ
θ θθθ
��
��
��
��
��
��
��
��
==
==
==
==
==
==
��
��
��
��
��
��
��
��
− −−−�
��
��
��
��
��
��
��
�
′ ′′′�
��
��
��
��
��
��
��
�=
==
==
==
=�
��
��
��
��
��
��
��
�′ ′′′
��
��
��
��
��
��
��
��
�
�
�
�
′ ′′′=
=+
−+
==
+−
+=
=+
−+
==
+−
+
�
�
�
�− −−−
�
�
�
�
�
3
5
Her
e 2 2
1ρ ρρρ
= === a
lwa
ys,
an
d
2 11
(1)
ρ ρρρ=
−=
−=
−=
− i
f 1
0(
0)
m′ ′′′
≥<
≥<
≥<
≥<
.
Th
e d
iag
on
alized
Dir
ac
-Majo
ran
a m
as
s t
erm
:
11
()
,2
2
.
DM
cc
kkL
kL
kL
kL
kk
k
kk
cc
kkL
kL
k
Lm
mν
νν
νν
νν
νν
νν
νν
νν
νν
νν
νν
νν
ν
νν
νν
νν
νν
νν
νν
νν
νν
+ +++=
−+
=−
=−
+=
−=
−+
=−
=−
+=
−
=+
==
+=
=+
==
+=
��
��
��
��
Th
ere
fore
, th
e t
wo
mas
siv
e n
eu
trin
os a
re M
ajo
ran
a p
art
icle
s.
Th
e “
sees
aw
” m
ech
an
ism
for
neu
trin
o m
ass
es
Fro
m e
xp
erim
enta
l d
ata
we
kn
ow
th
at
,.
qm
mm
ν ννν�
�
In o
rder
to s
up
pre
ss n
eutr
ino
ma
sses
let
us
ass
um
e th
at
bey
on
d t
he
SM
(a
t
ult
ra-h
igh
en
ergy
) th
ere
exis
ts a
mec
han
ism
gen
erati
ng t
he
righ
t-h
an
ded
3
6
Ma
jora
na m
ass
ter
m (
RD
mm
�),
an
d t
he
Dir
ac
mass
ter
m i
s g
ener
ate
d w
ith
the
sta
nd
ard
Hig
gs
mec
ha
nis
m. T
he
sees
aw
mec
ha
nis
m [
M. G
ell-
Ma
nn
, P
.
Ram
on
, R
. S
lan
sky
(1979
), T
. Y
an
agid
a (
1979
); R
. N
. M
oh
ap
atr
a a
nd
G.
Sen
jan
ov
ic (
19
80
)] i
s b
ase
d o
n t
he
Dir
ac-
Ma
jora
na
mass
ma
trix
wit
h
0L
m= ===
(it
is a
ssu
med
no H
igg
s tr
iple
ts):
0
.D
DR
mM
mm
��
��
��
��
= ===�
��
��
��
��
��
��
��
�
Her
e
or
.D
qm
mm
��
Th
e n
eutr
ino
Rν ννν
is
com
ple
tely
neu
tra
l u
nd
er t
he
Sta
nd
ard
-mo
del
gau
ge
gro
up
(2)
(1)
LY
SU
U× ×××
, an
d
Rm
is
no
t co
nn
ecte
d w
ith
th
e S
M s
ym
met
ry b
rea
kin
g
sca
le
( ((() )))1
/2
v2
24
6G
eVF
G− −−−
= ===�
,
bu
t is
ass
oci
ate
d t
o a
dif
fere
nt
hig
her
mass
sca
le, e.
g., t
he
GU
T-s
cale
:
.R
GU
TD
mM
m�
�
3
7
Dia
gon
ali
zati
on
of
the
ma
ss m
atr
ix M
giv
es
21
12
2
12
12
0,
,;
0
cos
sin
,
sin
cos
,
tan
2/
1.
TD
DR
D
R
LL
L
c RL
L
DR
mm
UM
Um
mm
mm
mm
i
i
mm
νν
θν
θν
νθ
νθ
νν
θν
θν
νθ
νθ
νν
θν
θν
νθ
νθ
νν
θν
θν
νθ
νθ
θ θθθ
��
��
��
��
= ===�
��
��
��
��
��
��
��
�
=+
=+
=+
=+
=−
+=
−+
=−
+=
−+
= ===
��
��
�
In t
he
case
of
thre
e g
ener
ati
on
s,
1 2 3
0,
=,
,
c
eLs
RT
cc
D
LL
Rs
R
cD
R
Ls
R
MM
MM
µ µµµ τ τττ
νν
νν
νν
νν
νν
νν
νν
νν
νν
νν
νν
νν
νν
νν
νν
νν�
��
��
��
��
��
��
��
��
��
��
��
��
��
��
��
��
��
��
��
�=
==
==
==
=�
��
��
��
��
��
��
��
��
��
��
��
��
��
��
��
��
��
��
��
��
��
��
��
��
��
��
��
��
��
��
��
�
the
sees
aw
mec
ha
nis
m l
ead
s to
a m
ass
sp
ectr
um
of
Majo
ran
a n
eutr
inos
{ {{{} }}}
km
wit
h 3
lig
ht
ma
sses
(1,2
,3k
= ===)
an
d 3
ver
y h
eavy
on
es (
4,5
,6k
= ===):
3
8
16
6
1
dia
g(
,,
);
.2
T DM
k
kk
k
VM
Vm
m
mL
νν
νν
νν
νν
+ +++
= ===
= ===
=−
=−
=−
=−� ���
�
ligh
t1
hea
vy
1
ligh
th
eavy
0,
O(
),0
,.
T
RD
T DR
DR
MV
WZ
WM
WM
MM
MM
MM
MM
− −−−
− −−−
��
��
��
��
==
+=
=+
==
+=
=+
��
��
��
��
��
��
��
��
��
��
��
��
=−
==
−=
=−
==
−=
T
he
mix
ing r
ela
tio
n
( ((() )))
( ((() )))
16
66
11
,,
,;
,,
,,
T
LL
LL
L
c
Lk
kL
sRsk
kL
kk
NV
nn
Ve
V
νν
νν
νν
νν
νν
µτ
νν
νν
µτ
νν
νν
µτ
νν
νν
µτ
νν
==
==
==
==
==
==
==
==
==
==
==
==
==
==
��
��
��
��
��
�
�
wh
ere
(4,5
,6)
kV
k= ===
� a
nd
(
1,2
,3)
skV
k= ===
are
ver
y s
ma
ll (
/D
Rm
m�
).
3
9
Th
e ch
arg
ed c
urr
ent
Lag
ran
gia
n i
n t
erm
s o
f th
e m
ass
ive
neu
trin
o f
ield
s k
ν ννν:
6
,,
1
H.c
.2
CC
Lk
kL
ek
gL
WU
α αααα ααα
µτ
µτ
µτ
µτ
γν
γν
γν
γν
+ +++
==
==
==
==
=−
+=
−+
=−
+=
−+
��
��
��
��
�
�
�
Her
e U
is t
he
36
× ××× n
eutr
ino
mix
ing m
atr
ix:
,
LU
AV
+ +++= ===
an
d
LA
is
the
33
× ××× u
nit
ary
ma
trix
ari
sin
g f
rom
dia
gon
ali
zati
on
of
the
Dir
ac
ma
ss t
erm
fo
r th
e ch
arg
ed l
epto
ns:
( ((() )))
00
,,
0
H.c
.H
.c.
,
(,
,),
;
dia
g,
,,
(,
),,
ml
Rl
LR
lL
e
T
LR
Rl
Ll
e
PP
PP
P
Ll
Ml
lD
lm
le
AM
AD
mm
m
lA
lP
LR
AA
I
µτ
µτ
µτ
µτ
µτ
µτ
µτ
µτ
µτ
µτ
µτ
µτ
= ===
+ +++
+ +++
=−
+=
−+
=−
=−
+=
−+
=−
=−
+=
−+
=−
=−
+=
−+
=−
==
+=
=+
==
+=
=+
==
==
==
==
==
==
==
==
==
==
� ����
�
��
���
wh
ere
0 l a
nd
l a
re w
eak
an
d m
ass
eig
enst
ate
s, r
esp
ecti
vel
y.
In g
ener
al
case
th
e n
um
ber
s
n o
f el
ectr
ow
eak
-sin
gle
t (“
ster
ile”
) n
eutr
ino
s sR
ν ννν
is a
rbit
rary
, an
d t
he
sees
aw
mec
ha
nis
m y
ield
s a s
et o
f 3
lig
ht
ma
sses
an
d
sn
4
0
larg
e m
ass
es. T
her
e ex
ists
a l
arg
e n
um
ber
of
sees
aw
mod
els
in w
hic
h b
oth
Dm
an
d
Rm
va
ry o
ver
ma
ny o
rder
s of
ma
gn
itu
de,
wit
h
Rm
ran
gin
g s
om
ewh
ere
bet
wee
n t
he
TeV
sca
le a
nd
th
e G
UT
sca
le (
15
16
10
10
GeV
÷ ÷÷÷�
).
Fo
r ex
am
ple
, fr
om
th
e a
tmosp
her
ic n
eutr
ino
osc
illa
tio
ns,
2
2
3atm
510
eVa
mm
m− −−−
>∆
≡×
>∆
≡×
>∆
≡×
>∆
≡×
�.
Ass
um
ing
2
,D
a
R
mm
m= ===
we
ob
tain
2
15
0.6
10
GeV
for
174
GeV
,D
RD
t
a
mm
mm
m=
×=
=×
==
×=
=×
=�
�
an
d
5T
eVfo
r 0
.511
MeV
.R
De
mm
m= ===
��
C
on
sid
er a
poss
ible
sce
nari
o o
f th
e g
ener
ati
on
of
the
Dir
ac-
Majo
ran
a m
ass
term
su
itab
le f
or
the
sees
aw
mec
han
ism
. T
he
gra
nd
un
ifie
d g
rou
p
(10
)S
O c
an
bre
ak
to
th
e S
M g
rou
p
(3)
(2)
(1)
SM
cL
YG
SU
SU
U=
××
=×
×=
××
=×
× t
hro
ugh
th
e ch
ain
4
1
( ((() )))
Vv
(10
)(1
)3
(1)
,G
UT
SM
BL
SM
emc
SO
GU
GS
UU
Λ ΛΛΛ
− −−−
→
×
→
→×
→×
→
→
×
→
×
→
→×
→×
→
→
×
ind
uci
ng t
he
ma
ss t
erm
1
H.c
.,2
DM
c
LD
RR
RR
LM
Mν
νν
νν
νν
νν
νν
νν
νν
ν+ +++
=−
−+
=−
−+
=−
−+
=−
−+
wh
ere
DM
is
a 3
sn
× ××× D
ira
c m
ass
matr
ix a
nd
R
M i
s a
ss
nn
× ××× M
ajo
ran
a m
ass
ma
trix
:
v/
2,
V/
2,
DR
My
MY
==
==
==
==
y a
nd
Y a
re m
atr
ices
of
Yu
ka
wa
co
up
lin
gs.
Th
e b
rea
kin
g s
cale
s:
( (((
) )))1/2
15
16
10
10
GeV
, V
11
0 T
eV,
v2
24
6 G
eV.
GU
TF
G− −−−
Λ÷
÷=
Λ÷
÷=
Λ÷
÷=
Λ÷
÷=
��
�
D
iago
na
liza
tion
of
the
neu
trin
o m
ass
ma
trix
by m
ean
s o
f a
un
ita
ry
( ((() )))
( ((() )))
33
ss
nn
+×
++
×+
+×
++
×+
ma
trix
V g
ives
3 l
igh
t an
d
sn
hea
vy M
ajo
ran
a n
eutr
ino
s:
3
3li
gh
th
eavy
14
sn
Lk
kL
kkL
kk
VV
νν
νν
νν
νν
νν
νν
+ +++
==
==
==
==
=+
=+
=+
=+
��
��
��
��
��
�.
4
2
Co
ncl
usi
on
N
eutr
ino
ph
ysi
cs i
s a b
oom
ing
fie
ld o
f re
sea
rch
in
vo
lved
wit
h p
art
icle
ph
ysi
cs, n
ucl
ear
ph
ysi
cs, a
stro
ph
ysi
cs, an
d c
osm
olo
gy
. In
gen
era
l, n
eutr
ino
s
pla
y a
sp
ecia
l ro
le i
n p
hysi
cs d
ue
to t
hei
r in
tim
ate
con
nec
tio
n w
ith
fu
nd
am
enta
l
sym
met
ries
an
d c
on
serv
ati
on
la
ws:
� e
ner
gy a
nd
mom
entu
m,
� l
epto
n n
um
ber
,
� p
ari
ty a
nd
ch
arg
e co
nju
gati
on
,
� C
P
inva
rian
ce,
� C
PT
an
d L
ore
ntz
in
va
ria
nce
.
T
he
rece
nt
dis
cov
ery
of
neu
trin
o o
scil
lati
on
s h
as
op
ened
ph
ysi
cs o
f m
ass
ive
an
d m
ixed
neu
trin
os.
Th
ere
is a
gen
eral
con
sen
sus
tha
t n
eutr
ino
ma
sses
hav
e
thei
r o
rig
in i
n a
New
Ph
ysi
cs b
eyo
nd
th
e S
M.
T
he
dee
per
un
der
sta
nd
ing
of
the
neu
trin
o p
rop
erti
es r
ema
ins
an
op
en f
un
dam
enta
l p
roble
m.
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