Dirac and Majorana Neutrino Masses
Carlo Giunti
INFN, Sezione di Torino, and
Dipartimento di Fisica Teorica, Università di Torino
Neutrino Unbound: http://www.nu.to.infn.it
La Massa dei Neutrini, Padova, 4-6 May 2010
C. Giunti and C.W. Kim
Fundamentals of Neutrino Physics
and Astrophysics
Oxford University Press
15 March 2007 – 728 pages
C. Giunti
Dirac and Majorana Neutrino Masses
5 May 2010
1
Fermion Mass Spectrum
1012
t
1011
1010
b
c
109
10
107
m [eV]
s
µ
8
106
τ
ντ
d
u
e
νµ
105
104
103
102
10
νe
1
10−1
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Dirac and Majorana Neutrino Masses
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2
Dirac Lepton Masses
LL
L
`L
!
`R
R
Lepton-Higgs Yukawa Lagrangian
y ` LL Φ `R
LD =
e R + H.c.
y LL Φ
Symmetry Breaking
1
Φ(x) = p
2
L
D
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0
v
=
!
e = i 2 Φ = p1
Φ
2
` py L
2
`L
py L
2
`L
0!
v
v!
0
Dirac and Majorana Neutrino Masses
v
0
`R
R + H.c.
5 May 2010
3
!
LD =
v
y ` p `L `R
2
m` = y `
pv
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2
v
y p L R + H.c.
2
m = y Dirac and Majorana Neutrino Masses
pv
2
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4
Three-Generations Dirac Neutrino Masses
0
0
eL
L0eL `0eL eL0
0
L0L 1
A
1
0 L A
`0L L0
`0R R0
0 R
`0eR eR0
0
eR
0
0 L
L0 L `0 L L0
`0 R R0
0 R
Lepton-Higgs Yukawa Lagrangian
LD =
h
X
; =e ;;
Symmetry Breaking
0 1
0
1
Φ(x) = p A
2 v
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i
0` L0 Φ `0 + Y 0 L0 Φ
e 0
Y
L R + H.c.
R
L
0 1
e = i 2 Φ = p1 v A
Φ
2 0
Dirac and Majorana Neutrino Masses
5 May 2010
5
1
A
L
D
X
0` `0 `0 + pv Y 0 0 0 + H.c.
pv Y
L R
L R
2
2
; =e ;;
=
L
D
=
0 01
e
B LC
ℓ0L 0L A
v
ℓ0L M 0` ℓ0R + p νL0 M 0 νR0 + H.c.
2
0 01
e
B R0 C
ℓ0R R
A
L0
R0
M 0` =
pv
2
Y 0`
1
0 0`
Mee Me0` Me0`
B
0` M 0` C
M 0` M0`e M
A
M0`e
0` M
0`
M
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0 0 1
B eL C
νL0 0 L A
0 0 1
B eR C
νR0 0 R A
0 L
M 0 =
0 R
pv
2
Y 0
1
0 0
Mee Me0 Me0
B
0 M 0 C
M 0 M0e M
A
Dirac and Majorana Neutrino Masses
M0e
5 May 2010
0
M
6
0
M
LD =
ℓ0L M 0` ℓ0R
νL0 M 0 νR0 + H.c.
Diagonalization of M 0` and M 0 with unitary VL` , VR` , VL , VR
ℓ0L = VL` ℓL
ℓ0R = VR` ℓR
νL0 = VL nL
νR0 = VR nR
Kinetic terms are invariant under unitary transformations of the fields
LD =
`y
ℓL VL M 0` VR` ℓR
y
νL VL M 0 VR νR + H.c.
`y
VL M 0` VR` = M `
` = m` Æ
M
(; = e ; ; )
y
VL M 0 VR = M = m Æ
Mkj
k kj
(k ; j = 1; 2; 3)
Real and Positive m` , mk
C. Giunti
Dirac and Majorana Neutrino Masses
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7
Massive Chiral Lepton Fields
0 1
eL C
B
C
B
`y ℓ0
C
ℓL = V L L B
B L C
A
0 1
eR C
B
C
B
`y ℓ0
C
ℓR = V R R B
BR C
A
0 L1
1L C
B
B
C
y
0
nL = VL νL B
B2L C
C
A
0 R 1
1R C
B
B
C
y
0
nR = VR νR B
B2R C
C
A
3L
LD =
=
ℓL M ` ℓR
X
=e ;;
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3R
nL M nR + H.c.
m` `L `R
3
X
mk kL kR + H.c.
k=1
Dirac and Majorana Neutrino Masses
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8
Mixing
Charged-Current Weak Interaction Lagrangian
(CC)
LI
=
pg
2 2
Weak Charged Current:
jW W + H.c.
jW = jW ;L + jW ;Q
Leptonic Weak Charged Current
X 0 0
jW ;L = 2
L `L = 2 νL0 ℓ0L
=e ;;
ℓ0L = VL` ℓL
νL0 = VL nL
y
y
jW ;L = 2 nL VL VL` ℓL = 2 nL VL VL` ℓL = 2 nL U y ℓL
Mixing Matrix
y
U y = VL VL`
C. Giunti
`y
U = VL VL
Dirac and Majorana Neutrino Masses
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◮
Definition: Left-Handed Flavor Neutrino Fields
0 1
eL
B C
`y
νL = U nL = VL νL0 = L A
L
◮
They allow us to write the Leptonic Weak Charged Current as in the SM:
X
jW ;L = 2 νL ℓL = 2
L `L
=e ;;
◮
Each left-handed flavor neutrino field is associated with the
corresponding charged lepton field which describes a massive charged
lepton:
jW ;L = 2 (eL eL + L L + L L )
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Dirac and Majorana Neutrino Masses
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10
Flavor Lepton Numbers
Flavor Neutrino Fields are useful for defining
Flavor Lepton Numbers
as in the SM
Le
L
L
+1
0
0
)
0
+1
0
)
0
0
+1
(e ; e )
( ;
( ; (ec ; e + )
c ; +
(c ; + )
Le
L
L
1
0
0
0
0
1
0
0
1
L = Le + L + L
Standard Model:
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Lepton numbers are conserved
Dirac and Majorana Neutrino Masses
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11
LD =
eL L
10 1
0 D
D
D
B mee me me C B eR C
D
D
L mDe m
m
A R A + H.c.
mDe
D
m
D
m
R
Le , L , L are not conserved
L is conserved:
C. Giunti
L(R ) = L( L )
Dirac and Majorana Neutrino Masses
) j∆Lj = 0
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12
Mixing Matrix
◮
0
1
Ue1 Ue2 Ue3
B
C
`y
U = VL VL = U1 U2 U3 A
U 1 U 2 U 3
◮
Unitary N N matrix depends on N 2 independent real parameters
N=3
=)
N (N 1)
=3
2
N (N + 1)
=6
2
Mixing Angles
Phases
◮
Not all phases are physical observables
◮
Only physical effect of mixing matrix occurs through its presence in the
Leptonic Weak Charged Current
C. Giunti
Dirac and Majorana Neutrino Masses
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13
◮
Weak Charged Current: jW ;L = 2
3
X
X
k=1 =e ;;
kL U k `L
◮
Apart from the Weak Charged Current, the Lagrangian is invariant
under the global phase transformations (6 arbitrary phases)
k ! e i 'k k (k = 1; 2; 3) ;
` ! e i ' ` ( = e ; ; )
◮
Performing this transformation, the Charged Current becomes
3
X
X
jW ;L = 2
kL e i 'k U k e i ' `L
k=1 =e ;;
3
'e ) X X e i ('k '1 ) U e i (' 'e ) `
kL |
L
{z }
{z } k | {z }
k=1 =e ;;
1
2
2
There are 5 arbitrary phases of the fields that can be chosen to eliminate
5 of the 6 phases of the mixing matrix
5 and not 6 phases of the mixing matrix can be eliminated because a
common rephasing of all the fields leaves the Charged Current invariant
() conservation of Total Lepton Number.
jW ;L = 2 e|
◮
◮
i ('1
C. Giunti
Dirac and Majorana Neutrino Masses
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14
◮
The mixing matrix contains 1 Physical Phase.
◮
It is convenient to express the 3 3 unitary mixing matrix only in terms
of the four physical parameters:
3 Mixing Angles and 1 Phase
C. Giunti
Dirac and Majorana Neutrino Masses
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15
Majorana Mass
Dirac Mass Lagrangian
LD =
R
1 D
L
2
LM =
m
2
C. Giunti
!
m (R L + L R )
! LC = C L T
m
2
LT C y L + L C L T
Majorana Mass Lagrangian
LT C y L + L C L T
=
Dirac and Majorana Neutrino Masses
m C
2 L
5 May 2010
L + L LC
16
= L + LC
◮
Majorana Field:
◮
Majorana Condition:
◮
1
Majorana Lagrangian: L M = m
2
◮
◮
C
=
The factor 1=2 distinguishes the Majorana Lagrangian from the Dirac
Lagrangian
Common terminology:
Majorana neutrino with negative helicity neutrino
Majorana neutrino with positive helicity antineutrino
C. Giunti
Dirac and Majorana Neutrino Masses
5 May 2010
17
Lepton Number
Z Z
L
!
= C
Z
L=
+1
Z
=) L = +1
LM =
LC
m C
2 L
Z Z
Z
L=
Z1
=) L =
L + L LC
1
Total Lepton Number is not conserved:
∆L = 2
Best process to find violation of Total Lepton Number:
Neutrinoless Double- Decay
N (A; Z ) ! N (A; Z + 2) + 2e + H
2¯
H
e
+
N (A; Z ) ! N (A; Z 2) + 2e + H
2
H
e
C. Giunti
Dirac and Majorana Neutrino Masses
(0 )
(0+ )
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18
No Majorana Neutrino Mass in the SM
h
◮
◮
Eigenvalues of the weak isospin I , of its third component I3 , of the
hypercharge Y and of the charge Q of the lepton and Higgs multiplets:
lepton doublet
lepton singlet
Higgs doublet
◮
i
Majorana Mass Term / LT C y L L C L T involves only the neutrino
left-handed chiral field L , which is present in the SM (one for each
lepton generation)
0 1
L
LL = A
`L
1=2
`R
0
0
1
+ (x)A
Φ(x) = 1=2
0 (x)
LT C y L has I3 = 1 and Y
C. Giunti
I
=
I3
Y Q = I3 +
1=2
1
0
1=2
1=2
0
1
1=2
Y
2
2
1
1
+1
0
2 =) needed Higgs triplet with Y = 2
Dirac and Majorana Neutrino Masses
5 May 2010
19
Mixing of Three Majorana Neutrinos
◮
◮
1
L M = νL0T
2
0 0 1
B eL C
νL0 0 L A
0 L
=
C y M L νL0 + H.c.
1 X
L
0T C y M
0 L + H.c.
2 ; =e ;; L
In general, the matrix M L is a complex symmetric matrix
X 0T y L 0
L C M L =
X 0T L
L M (C y )T 0 L
;
X 0T y L 0
X
L
=
L C M L = 0TL C y M
0 L
;
;
L
L
M
= M
C. Giunti
;
()
ML = ML
Dirac and Majorana Neutrino Masses
T
5 May 2010
20
◮
◮
◮
◮
1
L M = νL0T
2
0
ν =V n
L
L
L
C y M L νL0 + H.c.
=)
(VL )T M L VL = M ;
Mkj = mk Ækj
C y M L VL νL0 + H.c.
(k ; j = 1; 2; 3)
0 1
1L
B
C
y
0
Left-handed chiral fields with definite mass: nL = VL νL = 2L A
LM =
◮
1
L M = νL0T (VL )T
2
3
X
1
mk
2 k=1
T y
kL
C kL kL C kLT
Majorana fields of massive neutrinos:
LM =
C. Giunti
k
3L
C
= kL + kL
kC
3
1X
mk k k
2 k=1
Dirac and Majorana Neutrino Masses
5 May 2010
21
= k
Mixing Matrix
◮
Leptonic Weak Charged Current:
jW ;L = 2 nL U y ℓL
◮
with
`y
U = VL VL
Definition of the left-handed flavor neutrino fields:
0 1
eL
B C
`y
νL = U nL = VL νL0 = L A
L
◮
Leptonic Weak Charged Current has the SM form
X
jW ;L = 2 νL ℓL = 2
L `L
=e ;;
◮
Important difference with respect to Dirac case:
Two additional CP-violating phases: Majorana phases
C. Giunti
Dirac and Majorana Neutrino Masses
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22
◮
3
1X
T y
mk kL
Majorana Mass Term L =
C kL + H.c. is not invariant
2 k=1
under the global U(1) gauge transformations
(k = 1; 2; 3)
! e i 'k M
kL
kL
◮
Left-handed massive neutrino fields cannot be rephased in order to
eliminate two Majorana phases factorized0on the right of1mixing matrix:
1 0
0
B
C
U = UD DM
D M = 0 e i 2
0 A
0 0 e i 3
◮
U D is analogous to a Dirac mixing matrix, with one Dirac phase
◮
Standard parameterization:
0
U=
s12 c23
s12 s23
c12 c13
c12 s23 s13 e i Æ13
c12 c23 s13 e i Æ13
C. Giunti
c12 c23
c12 s23
s12 c13
s12 s23 s13 e i Æ13
s12 c23 s13 e i Æ13
Dirac and Majorana Neutrino Masses
10
1
s13 e i Æ13
s23 c13 A 0
0
c23 c13
5 May 2010
23
0
e i 2
0
1
0
0 A
e i 3
One Generation Dirac-Majorana Mass Term
If
R
exists, the most general mass term is the
Dirac-Majorana Mass Term
L D+M = L D + L L + L R
LD =
LL =
LR =
mD R L + H.c.
1
mL LT
2
1
mR RT
2
C. Giunti
C y L + H.c.
C y R + H.c.
Dirac Mass Term
Majorana Mass Term
New Majorana Mass Term!
Dirac and Majorana Neutrino Masses
5 May 2010
24
◮
Column matrix of left-handed chiral fields: NL =
L D+M =
◮
◮
1 T
N
2 L
C y M NL + H.c.
Diagonalization: nL = U y NL = 1L
=
mL mD
mD mR
!
2L!
L D+M =
m1 0
0 m2
1 X
T y
mk kL
C kL + H.c. =
2 k=1;2
k
◮
!
L
C! R T
!
The Dirac-Majorana Mass Term has the structure of a Majorana Mass
Term for two chiral neutrino fields coupled by the Dirac mass
UT M U =
◮
M=
L
RC
C
= kL + kL
Massive neutrinos are Majorana!
C. Giunti
Real mk
0
1 X
mk k k
2 k=1;2
k = kC
Dirac and Majorana Neutrino Masses
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25
Real Mass Matrix
◮
◮
◮
CP is conserved if the mass matrix is real: M = M M=
mL mD
mD mR
!
we consider real and positive mR and mD and real mL
A real symmetric mass matrix
! with U = O ! can be diagonalized
sin #
= 01 0
O = cossin## cos
2k = 1
#
2
MO
!
2mD
m10 0
=
tan 2# =
0
0 m2
mR mL
q
1
2
m20 ;1 =
mL + mR (mL mR )2 + 4 mD
2
◮
O
◮
m10 is negative if mL mR
T
U MU = T
T
O
C. Giunti
T
< mD2
M O =
21 m10
0
0
2
2 m20
Dirac and Majorana Neutrino Masses
!
=)
5 May 2010
mk = 2k mk0
26
◮
m20 is always positive:
q
1
0
m2 = m2 =
mL + mR + (mL
2
◮
If mL mR
1 = 1 and 2 = 1
If mL mR
mR ) +
2
4 mD
mD2 , then m10 0 and 21 = 1
1
m1 =
mL + mR
2
◮
2
q
(mL
=)
1 = i
1
2
q
(mL
and
C. Giunti
2 = 1
mR ) +
U=
cos # sin #
sin # cos #
1
2
mR )2 + 4 mD
=)
2
4 mD
!
< mD2 , then m10 < 0 and 21 =
m1 =
2
(mL + mR )
!
U=
Dirac and Majorana Neutrino Masses
i cos # sin #
i sin # cos #
5 May 2010
27
Special cases:
=)
◮
mL = mR
◮
mL = mR = 0 =) Dirac Limit
◮
◮
Maximal Mixing
jmL j; mR mD =) Pseudo-Dirac Neutrinos
mL = 0 mD mR =) See-Saw Mechanism
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Dirac and Majorana Neutrino Masses
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Maximal Mixing
mL = mR
# = =4
(
21 = +1 ;
21 = 1 ;
m20 ;1 = mL mD
m1 = mL mD if
m1 = mD mL if
m2 = mL + mD
mL
mL
mL
mD
< mD
< mD
8
i >
>
< 1L = p L RC
2
1 >
>
: = p + C
2L
L
R
2
8
h
i
i
>
C
>
= p (L + R )
LC + RC
< 1 = 1L + 1L
2
i
1 h
>
C
>
: 2 = 2L + 2L = p (L + R ) + LC + RC
2
C. Giunti
Dirac and Majorana Neutrino Masses
5 May 2010
29
Dirac Limit
◮
◮
mL = mR = 0
21 = 1 ;
22 = +1 ;
The two Majorana fields 1 and 2
m20 ;1 = mD
=)
field:
◮
(
m1 = mD
m2 = mD
can be combined to give one Dirac
=p
A Dirac field
1
(i 1 + 2 ) = L + R
2
can always be split in two Majorana fields:
i
1 h
C + + C
2
!
C
i
1
=p
i p
+p
2
2
2
=
◮
C
+
p
2
!
=
p1 (i 1 + 2 )
2
A Dirac field is equivalent to two Majorana fields with the same mass
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Dirac and Majorana Neutrino Masses
5 May 2010
30
Pseudo-Dirac Neutrinos
jmL j ; mR mD
◮
m20 ;1
◮
m10
◮
The two massive Majorana neutrinos are almost degenerate in mass
◮
The best way to reveal pseudo-Dirac neutrinos are active-sterile neutrino
oscillations due to the small squared-mass difference
' mL +2 mR mD
<0
=)
21 =
1 =) m2;1
∆m2
◮
' mD mL +2 mR
' mD (mL + mR )
The oscillations occur with practically maximal mixing:
tan 2# =
C. Giunti
2mD
mR mL
1
=)
Dirac and Majorana Neutrino Masses
# ' =4
5 May 2010
31
See-Saw Mechanism
[Minkowski, PLB 67 (1977) 42; Yanagida (1979); Gell-Mann, Ramond, Slansky (1979); Mohapatra, Senjanovic, PRL 44 (1980) 912]
mR
is forbidden by SM symmetries =) mL = 0
mD . v 100 GeV is generated by SM Higgs Mechanism
mL = 0
mD
◮ LL
◮
(protected by SM symmetries)
◮
◮
mR is not protected by SM symmetries =) mR
m10
2
D
' m
mR
m20 ' mR
9
=
;
=)
8
< 2 = 1 ;
1
: 2
MGUT v
2
D
'm
mR
2 = +1 ; m2 ' mR
m1
◮
Natural explanation of smallness of neutrino masses
◮
Mixing angle is very small: tan 2# = 2
◮
1 is composed mainly of L : 1L '
◮
mD
mR
ν1
ν2
1
i L
2 is composed mainly of R : 2L ' RC
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Dirac and Majorana Neutrino Masses
5 May 2010
32
Three-Generation Mixing
L D+M = L D + L L + L R
L
D
NS
X
X
=
LL =
s=1 =e ;;
1 X
L
0 L + H.c.
0T C y M
2 ; =e ;; L
S
1 X
0T
2 s ;s 0 =1 sR
N
LR =
N0L
L
D+M
!
νL0
ν 0C
1
= N0LT
2
R
0 M D 0 + H.c.
sR
s L
C y MssR 0 s0 0 R + H.c.
0 0 1
B eL C
νL0 0 L A
0 L
C y M D+M N0L + H.c.
C. Giunti
0 0C 1
1R
C
B
0
C
νR ... A
N0CS R
M
Dirac and Majorana Neutrino Masses
D+M
T
ML MD
=
MD MR
5 May 2010
33
!
◮
Diagonalization of the Dirac-Majorana Mass Term =) massive
Majorana neutrinos
◮
See-Saw Mechanism =) right-handed neutrinos have large masses and
are decoupled from the low-energy phenomenology
◮
At low energy we have an effective mixing of three Majorana neutrinos
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34