Física de Neutrinos

Fı́sica de Neutrinos
A. Mondragón y E. Peinado
Escuela de Fı́sica Nuclear 2007
Contenido
Primera Parte
• Oscilaciones de los neutrinos entre estados del sabor
• Observaciones y Experimentos
• Breve recordatorio de la Teorı́a: Modelo Estándar
• Neutrinos de Dirac y Majorana
• El mecanismo del subibaja (seesaw)
Segunda Parte
• Neutrinos en la Extensión S3-invariante del Modelo Estándar
• Las corrientes neutras que cambian el sabor
1
NEUTRINOS
Masas, mezclas de sabores y oscilaciones de los neutrinos
1957 Pontecorvo discutió las oscilaciones ν ↔ ν̄
1962 Maki, Nakagawa y Sakata propusieron las ”Transiciones virtuales” νe ↔ νµ
1964-1968 Davies observó el déficit de νe solares
1998 Oscilaciones de los neutrinos νµ atmosféricos
(SuperKamiokande)
2000 Observación directa de ντ en el experimento DONUT
2001 SNO mide las oscilaciones de los neutrinos νe solares y el flujo total de neutrinos
provenientes del Sol
2003 KAMLAND mide los parámetros de la oscilación de los neutrinos νe solares y los de
la oscilación de los ν̄e de reactores
2003-2007 Se midieron las diferencias de los cuadrados de las masas ∆m2ν ,ν y los ángulos
i j
de mezcla en las oscilaciones de los neutrinos entre estados del sabor y se refinaron las cotas
sobre la suma de las masas de los neutrinos con las medidas cosmológicas de presición.
ESTA ES LA EVIDENCIA EXPERIMENTAL INCONTROVERTIBLE DE FISICA NUEVA
MAS ALLA DEL MODELO STANDARD !!
2
3
Oscilaciones del sabor
|να i ≈
να
W
∗
U
i αi |νi i
P
α =sabor
i =masa
Propagación de los eigenestados de la masa
+
|νi(τ )i = e−imiτi |νi(0)i ≈ e−i(Eit−pi x)|νi(0)i
lα+
|να(L)i ≈
m2
∗ −i 2Ei L
|νi i
i Uαi e
P
Propagación de los estados del sabor
|να (L) >≈
XhX
β
i
i
m2
∗ − iL
Uαi e 2E Uβi |νβ
>
La probabilidad de transición está dada por
P (να → νβ ) = δαβ − 4
+2
h“
”
˜i
ˆ
∗
∗
2 L
2
Uαi Uβi Uαj Uβj sin 1.27∆mij ( E )
i>j Re
P
i
h“
”
∗
∗
2 L
Uαi Uβi Uαj Uβj sin[2.54∆mij ( E ]
i>j Im
P
4
5
6
7
8
9
10
11
12
13
14
Grupo de Norma del Modelo Standard
SUc(3) ⊗ SUW (2) ⊗ UY (1)
⇑
⇑
fuerte
débil
⇑
electromagnética
Los campos fermiónicos se describen por sus componentes de quiralidad izquierda o derecha
ψL,R =
1
(1 ± γ5)ψ,
2
ψ̄L,R = ψ̄(1 ± γ5)
I
1
2
I
QLi(3, 2)+1/6 , uRi (3, 1)+2/3 , dRi (3, 1)−1/3, LLi (1, 2), lRi (1, 1)
sabor, color, isospin débil, hipercarga
En el Modelo Standard hay un campo escalar llamado el campo de Higgs
φ(1, 2)+1/2
15
Interacciones de Norma del Modelo Standard
GM S = SUc (3) ⊗ SUW (2) ⊗ UY (1)
⇑
⇑
fuerte (color)
⇑
débil
electromagnética
La lagrangiana del Modelo Standard
L = Lkn + Lkf + LY + LH
Invariancia de Norma: ψ̂(x) → ψ̂ ′(x) = U (x)ψ̂(x)U + (x)
µ
µ
µ
µ
′
U (x) ǫ GM S
µ
∂ → D + igs Ga La + igAb Tb + ig B Y
Gµa(x) son los 8 campos de los gluones
Aµb(x) son los 3 bosones intermediarios de la interaccions débil
B µ(x) es el bosón de la hipercarga
16
La Lagrangiana del Modelo Standard
L = Lkn + Lkf + LY + LH
Energı́a cinética de los campos de Norma:
1
1
1
µν
µν
µν
Lkn = − Fµν F − Bµν B − Gµν G
4
4
4
a
a
a
Fµν = ∂µAν − ∂ν Aµ + gǫ
ijk
j
k
Aµ Aν
Bµν = ∂µBν − ∂ν Bµ
a
a
a
Gµν = ∂µGaµν − ∂ν Gµ + gs f
ijk
j
k
GµCν
17
La Lagrangiana de los fermiones
Lkf
X ¯+ µ
ˆψ γ Dµψ̂ +
=
i
i
i
µ
µ
µ
µ
′
µ
D = ∂ + igs Ga La + igAb Tb + ig B (x)Y
Lagrangiana del campo de Higgs
µ
†
+
LH = (D φ) (Dµ φ) − V (φ φ)
o
1 n
α
φ(x) = √ h(x)12×2 + iwα (x)σ
2
µ
D φ(x) =
“
”
i
i
′
~ µ − g Bµ φ
τ ·A
∂µ − g~
2
2
2
+
+
2
V (φ) = −µ φ φ + λ(φ φ)
18
Lagrangiana de Yukawa
3 “
X
−LY =
i,j=1
(ℓ) 1 − σ3
ν̄ℓi , ℓ̄i Γij φ
L
2
3 “
X
i,j=1
3 “
X
i,j=1
„ «
νℓj
+
ℓj R
”
(u) 1 + σ3
ūi , d¯i Γij
L
2
”
(d) 1 − σ3
ūi d¯i Γij
L
2
”
„
uj
dj
„ «
uj
+
dj R
«
+ c.h.
R
Γij son matrices complejas y hermitianas.
Conjugación de carga (C) y paridad (P)
CP :
bajo CP:
∗
+
ψ̄LiφψRj ←→ ψ̄Rj φ ψLi
+
∗
+
Γij ψ̄Li φψRj + Γij ψ̄Rj φ ψLi ←→ (Γij ψ̄Li φψRj + Γij ψ̄Rj φ ψLi )
LY invariante bajo CP si y sólo si Γij = Γ∗ij
Violación de CP en el Modelo Standard !!
n
o
(f )
∗(f )
d d†
†
Γij 6= Γij
y
Im det[Γ Γ , ΓuΓu] 6= 0
19
Masas de los fermiones
Mecanismo de Higgs: Cuando se rompe espontáneamente la simetrı́a de norma
< 0|φ(x)|0 >= v 6= 0
las interacciones de Yukawa generan las masas de los fermiones
I
l I
LM ass = d¯ILiMdij dIRj + ūILi Muij uIRj + l̄Li
Mij lRj + h.c.
Matrices de masas:
1
(f )
(f )
Mij = √ v Γij
2
Violación de CP
CP se viola si y sólo si
(f )
(f )∗
Mij 6= Mij
Si las matrices de masas son hermitianas
n
h
io
d
u
Im det M , M
6= 0 ⇔ CP
20
Los Neutrinos en el Modelo Estándar
En el Modelo Estándar
• Las masas de los quarks y los leptones cargados se generan en los sectores de Higgs y de Yukawa
• Los neutrinos no tienen masa
• El sector de Yukawa tiene demasiados parámetros libres (13)
Por lo tanto, debemos extender la teorı́a, con el objetivo de
• Dar masa a los neutrinos
• Sistematizar la fenomenologı́a tan rica de las masas y mezclas de los fermiones
• Reducir el número de parámetros
En las Extensiones del Modelo Standard
• Extendimos el concepto del sabor y generación al sector de Higgs
• Introdujimos una simetrı́a permutacional del sabor en el sector de las masas
• Generamos las masas de los neutrinos con el mecanismo del subibaja
21
Neutrinos de Dirac y Neutrinos de Majorana I
Espinores de Dirac
µ
iγ ∂µψ − mψ = 0.
Lagrangiana de Dirac
λ
LD = iψ̄γ ∂λψ − mψ̄ψ
Relación de energı́a y momento
λ
2
p pλ = m
ψ es un espinor de cuatro componentes
γ µ son las matrices de Dirac
Neutrinos de Dirac 6= Antineutrinos de Dirac
22
Neutrinos de Dirac y Neutrinos de Majorana II
Espinores de Majorana
La carga eléctrica de los neutrinos es nula
Si el campo espinorial = al campo conjugado de carga
c
ψM = ψM = C ψ̄
T
C es la matriz de conjugación de carga
ψM =
„
ν
iσ2ν ∗
«
ψM es un espinor de Majorana y tiene sólo dos componentes
Neutrinos de Majorana = Antineutrinos de Majorana
23
Las extensiones del Modelo Standard permiten Mν 6= 0
I Extender el sector de leptones agregando la parte derecha de los neutrinos
II Extender tanto el sector leptónico como el sector de Higgs
Neutrinos estériles:
Se agregan m campos fermiónicos neutros en singletes de SU (2)W y de quiralidad derecha.
Q = T3 + Y = 0
⇒
−LN =
νsI
i
“
”
1, 1
0
8
< Singletes de SU (3)C
Singletes de SU (2)L
:
hipercarga cero
1 c
ν
N̄i MNij Nj + Γij L̄Li φ̃Nj + h.c.
2
Después del rompimiento espontáneo de la simetrı́a de norma
`
−LMν = ν̄Li MD
´
ν
ij sj
´
1 c`
+ ν̄s MR ij νsj + h.c.
2 i
24
El Mecanismo del Subibaja (seesaw)
El mecanismo del subibaja
~
ν=
νL
I
νsj
!
i = 1, 2, 3; j = 4, 5, ....m
−LMν =
`
~
νL
´
ν̄sc
MN =
si MR ∼
<φ>
ΛF N
„
„
0
T
MD
0
T
MD
MD
MR
MD
MR
«„
c
νL
νs
«
«
<< 1
Ecuaciones de movimiento:
δLMν
−1
−1 T T
T
y νR = −MR
MD ν̄L )
δνR = 0 → (νR = −ν̄L MD MR
se obtiene LMν = − 12 ν̄LMcν νLc + h.c.
Mν = −MD MR−1MD .
25
26
Contents
• Flavour permutational symmetry
• A minimal S3 -invariant extension of the Standard Model
• Masses and mixings in the leptonic sector
• The neutrino mass spectrum
• FCNCs
• Summary and conclusions
27
Flavour permutational symmetry
• Prior to the introduction of the Higgs boson and mass terms, the Langrangian of the Standard
Model is chiral and invariant with respect to any permutation of the left and right quark fields.
GF ∼ S3L ⊗ S3R
• Charged currents Jµ are invariant under GF if the d and u−type fields are transformed with
the same family group matrix
Jµ = −iūLγµdL + h.c. ⇒ GF ∼ S3 ⊂ S3L ⊗ S3R
• When < 0|ΦH |0 >6= 0, the Yukawa couplings give mass to quarks and leptons, if we assume
that the S3 permutational symmetry is not broken
0
1
1 1 1
1
Mq = m3q @1 1 1A
3
1 1 1
mt 6= 0, me = mµ = 0; mb 6= 0, ms = md = 0
mτ 6= 0, mµ = me = 0; mντ = mνµ = mνe = 0
V=1
There is no mixing nor CP- violation.
28
The Group S3
The group S3 of permutations of three objects
Permutations
Rotations
3
„
1
3
V1
2
1
3
2
«
⇐⇒
a 120◦− rotation around the
invariant vector V1
V2A
2
„
1
2
2
1
3
3
«
⇐⇒
a 180◦rotation around the
invariant vector V2s
V2s
1
Symmetry adapted basis
|v2A
0 1
0 1
1
1
1
1
1
1
1
>= √ @−1A , |v2s >= √ @ 1 A , |v1 >= √ @1A
3 1
2
6 −2
0
0
29
Irreducible representations of S3
The group S3 has two one-dimensional irreps (singlets ) and one two-dimensional irrep (doublet)
• one dimensional: 1A antisymmetric singlet, 1s symmetric singlet
• Two - dimensional: 2 doublet Direct product of irreps of S3
1s ⊗ 1s = 1s ,
1s ⊗ 1A = 1A ,
1A ⊗ 1A = 1s ,
1s ⊗ 2 = 2,
1A ⊗ 2 = 2
2 ⊗ 2 = 1s ⊕ 1A ⊕ 2
the direct (tensor) product of two doublets
«
„
pD1
pD =
and
pD2
qD
«
„
qD1
=
qD2
T
has two singlets, rs and rA, and one doublet rD
rs = pD1qD1 + pD2qD2
is invariant,
T
rD
=
„
rA = pD1 qD2 − pD2 qD1
pD1 qD2 + pD2qD1
pD1qD1 − pD2qD2
is not invariant
«
30
A Minimal, S3 invariant extension of the SM
The Higgs sector is modified,
Φ→H
=
T
(Φ1, Φ2, Φ3 )
H is a reducible 1s ⊕ 2 rep. of S3
Hs
HD
=
=
”
1 “
√ Φ1 + Φ2 + Φ3
3
0
B
@
√1 (Φ1
2
− Φ2 )
1
C
A
1
√ (Φ1 + Φ2 − 2Φ3 )
6
Quark, lepton and Higgs fields are
QT = (uL , dL), uR , dR ,
L† = (νL, eL), eR , νR,
H
All these fields have three species (flavours) and belong to a reducible 1⊕ 2 rep. of S3
31
The most general renormalizable Yukawa interactions
LY = LYD + LYu + LYE + LYν
Quarks
LYD
=
−Y1dQI HS dIR − Y3dQ3 HS d3R − Y2d[ QI κIJ H1dJR + QI ηIJ H2dJR ]
−
Y4 Q3HI dIR − Y5 QI HI d3R + h.c
d
LYU
d
=
−Y1uQI (iσ2)HS∗ uIR − Y3uQ3 (iσ2)HS∗ u3R
−
Y2u[ QI κIJ (iσ2)H1∗ uJR + ηQI ηIJ (iσ2 )H2∗uJR ]
−
Y4 Q3 (iσ2)HI uIR − Y5 QI (iσ2)HI u3R + h.c.,
u
∗
∗
u
Doublets carry indices I, J = 1, 2
„
0
κ=
1
«
1
0
y
η=
„
1
0
«
0
1
Singlets carry the index s or 3
32
Yukawa interaction II
Leptons
LYE
=
−Y1e LI HS eIR − Y3e L3HS e3R − Y2e [ LI κIJ H1 eJR + LI ηIJ H2 eJR ]
−
Y4 L3HI eIR − Y5 LI HI e3R + h.c.,
e
LYν
e
=
−Y1ν LI (iσ2)HS∗ νIR − Y3ν L3(iσ2 )HS∗ ν3R
−
Y2 [ LI κIJ (iσ2 )H1 νJR + LI ηIJ (iσ2 )H2 νJR ]
−
ν
∗
∗
Y4ν L3(iσ2 )HI∗νIR − Y5ν LI (iσ2 )HI∗ν3R + h.c.
I, J = 1, 2
Furthermore, the Majorana mass terms for the right handed neutrinos are
LM
=
T
T
−M1νIR CνIR − M3 ν3R Cν3R ,
C is the charge conjugation matrix.
33
The Higgs potential
V
V1
=
+
V2
=
V1 + V2
h
i
µ (H̄D1 HD1) + (H̄D2 HD2) + (H̄3 H3 )
2
i2
1 h
λ1 (H̄D1 HD1) + (H̄D2 HD2) + (H̄3 H3)
2
=
h
i
η1 (H̄3H3 ) (H̄D1 HD1 + H̄D2HD2)
where
1
HD2 = √ (Φ1 + Φ2 − 2Φ3)
6
1
HD1 = √ (Φ1 − Φ2),
2
H3
=
1
√ (Φ1 + Φ2 + Φ3)
3
34
Mass matrices
We will assume that
< HD1 >
=
< HD2 > 6= 0
< H3 > 6= 0
and
and
2
2
< H3 > + < HD1 > + < HD2 >
2
≈
“ 246
2
GeV
”2
Then, the Yukawa interactions yield mass matrices of the general form
0
µ1 + µ2
M = @ µ2
µ4
µ2
µ1 − µ2
µ4
1
µ5
µ5 A
µ3
The Majorana masses for νL are obtained from the see-saw mechanism
Mν = MνD M̃
−1
T
(MνD )
with
M̃ = diag(M1 , M1 , M3)
35
Mixing matrices
The mass matrices are diagonalized by unitary matrices
†
Ud(u,e)L
Md(u,e)Ud(u,e)R
“
=
diag md(u,e)ms(c,µ)mb(t,τ )
and
Uν† Mν Uν
=
“
diag mν1 , mν2 , mν3
The masses can be complex, and so, UeL is such that
†
MeMe† UeL
UeL
=
“
2
2
diag |me | , |mµ | , |mτ |
The quark mixing matrix is
VCKM
=
”
”
2
”
,
etc.
†
UdL
UuL
and, the neutrino mixing matrix is
VM N S
=
†
UeLUν
36
The leptonic sector
To achieve a further reduction of the number of parameters, in the leptonic sector, we introduce
an additional discrete Z2 symmetry
−
HI , ν3R
+
HS , L3 , LI , e3R , eIR , νIR
then,
e
e
ν
ν
Y1 = Y3 = Y1 = Y5 = 0
Hence, the leptonic mass matrices are
0 e
µ2
Me = @µe2
µe4
µe2
−µe2
µe4
1
µe5
µe5 A
0
and
MνD
µν2
= @µν2
µν4
0
µν2
−µν2
µν4
1
0
0A
µν3
37
Mass matrix of the charged leptons
The unitary matrix UeL is calculated from
†
2
2
†
=
diag(me , mµ, mτ )
2|µe2 |2
|µe5 |2
|µe5|2
2µ2 µ∗4
|µe5 |2
2|µe2 |2 + |µe5 |2
0
2µ∗2 µ4
0
2|µe4 |2
UeL MeMeLUeL
2
where
†
Me Me
=
0
B
B
B
B
B
@
+
1
C
C
C
C
C
A
• We reparametrize Me Me† in terms of its eigenvalues
• Then, we compute UeL as function of the mass eigenvalues of MeMe†
38
Reparametrization of the mass matrix I
From the invariants of Me Me† we obtain
†
2
2
2
T r(Me Me ) = me + mµ + mτ =
m2τ
h
“
”i
2
2
2
4|µ̃2| + 2 |µ̃4 | + |µ̃5 |
,
†
2
2
2
2
2
χ(Me Me ) = mτ (me + mµ) + me mµ =
4
4mτ
and
h
“
”
i
4
2
2
2
2
2
|µ̃2 | + |µ̃2| |µ̃4| + |µ̃5 | + |µ̃4| |µ̃5| ,
†
2
2
2
6
2
2
det(Me Me ) = me mµmτ = 4mτ |µ̃2 | |µ̃4 | |µ̃5 |
2
39
Reparametrization of the Mass Matrix II
Solving for the parameters |µ̃2 |2, |µ̃4 |2, and |µ̃5|2 , in terms of mµ/mτ and me /mτ we get,
|µ̃2|
2
|µ̃4,5| =
1
4
“
1−
2 2
2 (1−x )
m̃µ 1+x2
− 4β
2
=
m̃2
µ 1+x4
2 1+x2
(1)
+β
”
v
0
u
u“
”
u
2 )2 2
2 2
B
(1−x
1+x2
1u
2 (1−x ) +
∓ 4 t 1 − m̃2µ 1+x2
− 8m̃2e 1+x
+
8β
1
−
m̃
@
4
µ 1+x2
1
(1+x2 )2 C
x2
A
2β(1+x2 ) (1+x4 )2
1+ 2
m̃µ (1+x4 )
+ 16β 2
(2)
and
x = me/mµ ,
2
2
1 me mµ
β≈
2 m4τ
40
β is the smallest root of the cubic equation:
z 2
1
z
z2
1
2
β − (1 − 2y + 6 )β − (y − y − 4 + 7z − 12 2 )β −
2
y
4
y
y
3
1 z2
z3
3 z2
1
yz −
− 3 =0
+
2
8
2y
4y
y
41
The Mass Matrix of the charged leptons as function of its eigenvalues
The mass matrix of the charged leptons is
0
m̃µ
√1 √
B
2 1+x2
B
B
B
B
B
m̃
√1 √ µ
Me ≈ mτ B
B
2 1+x2
B
B
B
@ qm̃e (1+x2 ) iδe
e
1+x2 −m̃2
µ
m̃µ
√1 √
2 1+x2
m̃
− √1 √ µ
2
1+x2
m̃ (1+x2 ) iδe
q e
e
1+x2 −m̃2
µ
√1
2
r
1+x2 −m̃2
µ
C
1+x2
1
C
C
C
r
C
2
1+x2 −m̃µ C
1
√
C.
C
2
1+x2
C
C
C
A
0
(3)
This expression is accurate to order 10−9 in units of the τ mass
There are no free parameters in Me other than the Dirac Phase δ!!
42
The Unitary Matrix UeL
The unitary matrix UeL is calculated from
We find
` 2
2
†
2´
†
UeL Me MeLUeL = diag me , mµ, mτ
UeL = ΦeLOeL ,
and
0
OeL
ˆ
˜
ΦeL = diag 1, 1, eiδD
2
2
4
(1+2m̃2
µ +4x +m̃µ +2m̃e )
√1 x q
4
6
2
2
2
4
1+m̃2
µ +5x −m̃µ −m̃µ +m̃e +12x
B
B
B
B
2
B
(1+4x2 −m̃4
µ −2m̃e )
B − √1 x q
≈B
4
6
2
2
2
4
1+m̃2
B
µ +5x −m̃µ −m̃µ +m̃e +12x
B
B
q
B
2
2
2
2
1+2x2 −m̃2
@
µ −m̃e (1+m̃µ +x −2m̃e )
−q
2
4
6
2
2
4
1+m̃µ +5x −m̃µ −m̃µ +m̃e +12x
− √12 q
2
4
(1−2m̃2
µ +m̃µ −2m̃e )
√1
2
4
6
2
2
1−4m̃2
µ +x +6m̃µ −4m̃µ −5m̃e
4
(1−2m̃2
µ +m̃µ )
1
√ q
2 1−4m̃2 +x2 +6m̃4 −4m̃6 −5m̃2
µ
µ
µ
e
q
2 ) 1+2x2 −m̃2 −m̃2
−2
m̃
(1+x2 −m̃2
µ
e
e
µ
−x q
4
6
2
2
1−4m̃2
µ +x +6m̃µ −4m̃µ −5m̃e
1
√1
2
√
1+x2 m̃e m̃µ
q
1+x2 −m̃2
µ
(4)
43
C
C
C
C
C
C
C ,
C
C
C
C
A
The neutrino mass matrix I
The neutrino mass matrix is obtained from the see-saw mechanism
Mν
=
0
2(ρν2 )2
−1
T
MνD M̃ (MνD ) = @ 0
2ρν2 ρν4
0
2(ρν2 )2
0
1
2ρν2 ρν4
A
0
2(ρν4 )2 + (ρν3 )2
Mν is reparametrized in terms of its eigenvalues
0
mν3
B
Mν = B
0
@ q
(mν3 − mν1 )(mν2 − mν3 )e−iδν
0
mν3
0
q
1
−iδν
(mν3 − mν1 )(mν2 − mν3 )e
C
C.
0
A
−2iδν
(mν1 + mν2 − mν3 )e
(5)
44
Neutrino Mass Matrix II
The complex symmetric matrix Mν is diagonalized as
T
Uν M ν Uν
=
0
|mν1 |eiφ1−iφν
@
0
0
where
0
1
Uν = @ 0
0
0
1
0
0
0
eiδν
0
1B
B
B
AB
B
B
@
s
−
0
|mν2 |eiφ2−iφν
0
mν2 − mν3
mν2 − mν1
s 0
mν3 − mν1
mν2 − mν1
s
1
0
0 A
|mν3 |
mν3 − mν1
mν2 − mν1
0
s
mν2 − mν3
mν2 − mν1
1
0 C
C
C
1 C
C.
C
A
0
(6)
the mass eigenvalues, mν1 , mν2 and mν3 are, in general, complex numbers
45
All the phases in Mν except for one, can be absorbed in a rephasing of the fields
the phases φ1 and φ2 are fixed by the condition
|mν3 | sin φν = |mν2 | sin φ2 = |mν1 | sin φ1
and therefore
mν2 − mν3
mν3 − mν1
=
(∆m212 + ∆m213 + |mν3 |2 cos2 φν )1/2 − |mν3 || cos φν |
(∆m213 + |mν3 |2 cos2 φν )1/2 + |mν3 || cos φν |
.
(7)
46
The neutrino mixing matrix I
†
VPthM N S = UeL
Uν
The theoretical mixing matrix VPthM N S is
VPthM N S
=
×
0
O11 cos η + O31 sin ηeiδ
B
B
B
B −O12 cos η + O32 sin ηeiδ
B
@
O13 cos η − O33 sin ηeiδ
1
@ 0
0
0
0
eα
0
O11 sin η − O31 cos ηeiδ
−O21
−O12 sin η − O32 cos ηeiδ
O22
O13 sin η + O33 cos ηeiδ
O23
1
C
C
C
C
C
A
(8)
1
0
0 A
eiβ
where Oij are the absolute values of the elements of Oe
P DG
VP M N S
=
0
c12 c13
B
@ −s12 c23 − c12 s23 s13 eiδ
s12 s23 − c12 c23 s13 eiδ
s12 c13
c12 c23 − s12 s23 s13 eiδ
−c12 s23 − s12 c23 s13 eiδ
−iδ
s13 e
s23 c13
c23 c13
10
1
CB
A@ 0
0
0
eiα
0
1
0
C
0 A
eiβ
47
The parameters in VP M N S I
From our previous expressions, the parameters in VP M N S are
sin θ13 ≈
sin θ23 ≈
2
tan θ12
=
(1+4x2 −m̃4
µ)
√1 x q
,
2
2 −m̃4
1+m̃2
+5x
µ
µ
(9)
4
1−2m̃2
µ +m̃µ
1
√ q
.
2 1−4m̃2 +x2 +6m̃4
µ
µ
(∆m212 + ∆m213 + |mν3 |2 cos2 φν )1/2 − |mν3 || cos φν |
(∆m213 + |mν3 |2 cos2 φν )1/2 + |mν3 || cos φν |
.
48
The parameters in VM N S II
The Majorana phases are
sin 2α = sin(φ1 − φ2 ) =
|mν3 | sin φν
|mν1 ||mν2 |
×
«
„q
q
|mν2 |2 − |mν3 |2 sin2 φν + |mν1 |2 − |mν3 |2 sin2 φν
sin 2β = sin(φ1 − φν ) =
«
„
q
q
sin φν
|mν3 | 1 − sin2 φν + |mν1 |2 − |mν3 |2 sin2 φν .
|mν1 |
The mixing angles θ13 and θ23 depend only on the charged lepton masses and are in excellent
agreement with the experimental values
2
Experimental
Theoretical
exp
sin θ13 = 1.1 × 10
(sin θ13)
2
≤ 0.025
+0.06
sin θ23 = 0.5−0.05
2
−5
2
sin θ23 = 0.499
49
The neutrino mass spectrum I
In the present model, the experimental restriction
|∆m221 | < |∆m223 |
implies an inverted neutrino mass spectrum mν3 < mν1 , mν2
From our previous expressions
q
∆m213
1 − tan4 θ12 + r 2
,
|mν3 | =
p
p
2 cos φν tan θ12 1 + tan2 θ12 1 + tan2 θ12 + r 2
where r = ∆m221 /∆m223 .
The mass |mν3 | assumes its minimal value when sin φν = 0,
|mν3 | ≈
q
2
1 ∆m13
2 tan θ12
(1 − tan2 θ12)
50
Neutrino mass spectrum II
• We wrote the neutrino mass differences, mνi − mνj , in terms of the differences of the squared
masses ∆2ij = m2ν − m2ν and one of the neutrino masses, say mν3 .
i
j
• The mass mν2 was taken as a free parameter in the fitting of our formula for tan θ12 to the
experimental value
• with
2
−3
∆m13 = 2.6 × 10
eV
2
2
−5
∆m21 = 7.9 × 10
eV
2
and
tan θ12 = 0.667
we get
|mν3 | ≈ 0.022eV
=⇒ |mν2 | ≈ 0.056eV
and
|mν1 | ≈ 0.055eV
51
FCNC I
In the Standard Model the FCNC a tree level are suppressed by the GIM mechanism. Models
with more than one Higgs SU (2) doublet have tree level FCNC due to the interchange of scalar
fields. The mass matrix written in temrs of the Yukawa couplings is
e
E1
0
E2
0
MY = Yw H1 + Yw H2 ,
FCNC processes:
φ0 (k)
φ0 (k)
µ(p′ )
τ (p)
µ(p′ )
τ (p)
li
µ
li
γ
γ
µ
φ0 (k)
µ
φ0
τ
µ(p′ )
τ (p)
li
γ
Figure 1: The figure in the left contributes to the process τ − → 3µ. The tree figures in the right
contribute to the process τ → µγ .
52
FCNC II
The Yukawa matrices in the weak basis are
0
E1
Yw
and
B
B
B
B
mτ B
B
=
v1 B
B
B
B
@
0
YwE2
B
B
B
B
mτ B
B
=
v2 B
B
B
B
@
r
1+x2 −m̃2
µ
2
1+x
0
m̃µ
√1 √
2
1+x2
m̃
√1 √ µ
2
1+x2
0
0
0
0
m̃ (1+x2 )
iδe
q e
e
1+x2 −m̃2
µ
m̃µ
√1 √
2
1+x2
0
0
√1
2
0
m̃
− √12 √ µ 2
1+x
m̃ (1+x2 )
iδe
q e
e
1+x2 −m̃2
µ
0
√1
2
r
1+x2 −m̃2
µ
2
1+x
0
1
C
C
C
C
C
C
C
C
C
C
A
(10)
1
C
C
C
C
C
C.
C
C
C
C
A
(11)
53
FCNC III
The Yukawa matrices in the mass basis defined by
†
ỸmEI = UeL
YwEI UeR
E1
Ỹm
and
ỸmE2
0
− 12 m̃e
1
2x
1
2 m̃µ
− 12
− 21 m̃µ
1
2
−m̃e
1
2 m̃e
− 21 x
m̃µ
1
2 m̃µ
1
2
− 21 m̃µ x2
1
2 m̃µ
1
2
2m̃e
B
mτ B
B
≈
B −m̃µ
v1 B
@
2
1
m̃
x
µ
2
0
B
mτ B
B
≈
B
v2 B
@
1
C
C
C
C ,
C
A
m
1
C
C
C
C ,
C
A
m
54
Branching ratios
We define the partial branching ratio (only leptonic decays)
Γ(τ → µe+ e−)
Br(τ → µe e ) =
Γ(τ → eν ν̄) + Γ(τ → µν ν̄)
+ −
thus
Br(τ → µe+e−) ≈
Similar computations lead to
9
4
„
me mµ
m2τ
3α
Br(τ → eγ) ≈
8π
„
«2
mµ
MH
mτ
MH1,2
«4
!4
,
,
« „
«
mµ 2 mτ 4
,
mτ
MH
„
«
9
mµ 4
Br(τ → 3µ) ≈
,
64 MH
« „
«
„
me mµ 2 mτ 4
,
Br(µ → 3e) ≈ 18
2
m
MH
„ τ «4 „
«
mτ 4
27α me
.
Br(µ → eγ) ≈
64π mµ
MH
3α
Br(τ → µγ) ≈
128π
„
55
Numerical results
Table 1: Leptonic processes via FCNC
FCNC processes
Theoretical BR
τ → 3µ
τ → µe+ e−
τ → µγ
τ → eγ
µ → 3e
µ → eγ
8.43 × 10−14
3.15 × 10−17
9.24 × 10−15
5.22 × 10−16
2.53 × 10−16
1 × 10−19
Experimental
upper bound BR
2 × 10−7
2.7 × 10−7
6.8 × 10−8
1.1 × 10−11
1 × 10−12
1.2 × 10−11
References
B. Aubert et. al.
B. Aubert et. al.
B. Aubert et. al.
B. Aubert et. al.
U. Bellgardt et al.
M. L. Brooks et al.
procesos FCNC pequeños que sirven de intermediarios a las interacciones no-standard de los
leptones cargados y los neutrinos podrı́an tener importancia en la descripción teórica del colapso
gravitacional del núcleo y la generación de choque en la etapa explosiva de una supernova
56
Muon Anomalous Magnetic Moment
Models with more than one Higgs SU (2) doublet also contribute to the anomalous magnetic
moment of the muon due to the interchange of scalars
µ(p)
Yµτ
µ(p′)
H(k)
τ
τ
Yτµ
γ(q)
The Anomalous Magnetic Moment is
!
2
MH
Yµτ Yτ µ mµ mτ
log
−
δaµ =
2
16π 2 MH
m2τ
that is
given by
!
3
2
−12
δaµ = 2.6 × 10
which is well below the difference
SM
∆aµ = aexp
= (287 ± 91) × 10−11
µ − aµ
57
Conclusions I
• By introducing three SU (2)L Higgs doublet fields in the theory, we extended the concept of
flavour and generations to the Higss sector and formulated a minimal S3 −invariant Extension
of the Standard Model
• A definite structure of the Yukawa couplings is obtained which permits the calculation of mass
and mixing matrices for quarks and leptons in a unified way
• A further reduction of free parameters is achieved in the leptonic sector by introducing a Z2
symmetry.
• The three charged lepton masses, three Majorana masses of the left-handed neutrinos and
the three mixing angles and the 2 Majorana phases are computed in terms of only seven free
parameters, in agreement with all the experimental observations at this time
• The magnitudes of the three mixing angles, θ12 , θ23 , and θ13, are determined by an interplay
of the S3 × Z2 symmetry, the see saw mechanism and the lepton mass hierachy
58
Conclusions II
• The mixing angles θ23 and θ13 are insensitive to the values of the neutrino masses
• The solar mixing angle fixes the scale and origen of the mass spectrum
• The neutrino mass spectrum has an inverted mass hierarchy with the values
|mν2 | ≈ 0.056eV,
|mν1 | ≈ 0.055eV
|mν3 | ≈ 0.022eV,
• Numerical values for FCNC below experimental constraints
• FCNC processes considered are strongly suppressed by powers of the small mass ratios me /mτ
and mµ/mτ , and by the ratio
“
”4
mτ /MH1,2
• Could be important in astrophysical processes
59