Neutrinos and the
Flavour Problem
Part I. The Flavour Problem
Part II. Neutrino Phenomenology
Part III. Flavour in the Standard Model
Part IV. Origin of Neutrino Mass
Part V. Flavour Models
Part VI. Discrete Family Symmetry
Part VII. GUT Models
04/02/2010
Steve King, University of Warwick, Coventry
Part I.
The Flavour Problem
04/02/2010
Steve King, University of Warwick, Coventry
04/02/2010
u c t
d s b
up
down
charm
top
strange bottom
νe νμ ντ
e-neutrino μ-neutrino τ-neutrino
e μ τ
electron
muon tau
I
II III
Generations of
matter
Steve King, University of Warwick,
Coventry
Vertical
The Flavour
Problem
1. Why are there
three families of
quarks and
leptons?
Leptons Quarks
Horizontal
The Flavour Problem
2. What is the origin of quark and
lepton masses?
t
×10−3
d
ν1
< ×10
×10−3
e
−12
Horizontal
direction
ν2
×10
c
×10−1
s
μ
×10−4
×1
u
b
×10−2
×10
−2
−12
ν3
×10−11
τ×10
×10−1
−1
Vertical
direction
quark and
lepton
masses
Normal
Inverted
Absolute neutrino mass scale?
The Flavour Problem
3. Why is quark mixing so small?
Cabibbo
Kobayashi
Maskawa
VCKM
0
0 ⎞ ⎛ c13
⎛1
⎜
0
= ⎜⎜ 0 c23 s23 ⎟⎟ ⎜
⎜ 0 −s
c23 ⎟⎠ ⎜⎝ −s13e i δCP
23
⎝
0 s13e −i δCP ⎞ ⎛ c12 s12
⎟
1
0 ⎟ ⎜⎜ −s12 c12
0
c13 ⎟⎠ ⎜⎝ 0
0
0⎞
0 ⎟⎟
1 ⎟⎠
θ12 = 13° ± 0.1°
All these angles are pretty small – why?
θ 23 = 2.4° ± 0.1°
θ13 = 0.20 ± 0.05
°
04/02/2010
°
While the CP phase is quite large
δCP≈70o ± 5o
Steve King, University of Warwick, Coventry
The Flavour Problem
4. What is origin of quark CP violation?
Wolfenstein
≈
λ ≈ 0.226
A ≈ 0.81 ρ ≈ 0.13 η ≈ 0.35
VudVub* + VcdVcb* + VtdVtb* = 0
α ≈ 90o ± 4o
δCP≈ γ ≈70o ± 5o
04/02/2010
Steve King, University of Warwick, Coventry
The Flavour Problem
5. Why is lepton mixing so large?
⎛ν e ⎞
⎜ − ⎟
⎝ e ⎠L
⎛ν μ ⎞
⎜ − ⎟
⎝ μ ⎠L
⎛ν τ ⎞
⎜ − ⎟
⎝τ ⎠L
Standard Model
states
Neutrino mass
states
m3
Pontecorvo
Maki
Nakagawa
Sakata
m2
m1
0
0 ⎞ ⎛ . c13
0 s13e − iδ ⎞ ⎛. c12 s12
⎛ 1.
⎜
⎟
U PMNS = ⎜⎜ 0 c23 s23 ⎟⎟ ⎜ 0
1
0 ⎟ ⎜⎜ − s12 c12
.
iδ
⎜ 0. − s
⎟⎜−
⎜. 0
⎟
c
s
e
0
c
0
23
23 ⎠ ⎝ . 13
13 ⎠ ⎝.
⎝.
sij = sin θij
Reactor
Solar
Atmospheric
cij = cos θij
Oscillation phase
Majorana phases
04/02/2010
δ
α1 , α 2
0 ⎞ ⎛. eiα1 / 2
⎜
0 ⎟⎟ ⎜ 0
.
1 ⎟⎠ ⎜⎝. 0
0
eiα2 / 2
0
0⎞
⎟
0⎟
1 ⎟⎠
Majorana
3 masses + 3 angles + 1(3) phase(s)
= 7(9) new parameters for SM
Steve King, University of Warwick, Coventry
Part II.
Neutrino
Phenomenology
04/02/2010
Steve King, University of Warwick, Coventry
Global Fit to Atmospheric and Solar Data
Atmospheric
Solar
04/02/2010
Steve King, University of Warwick, Coventry
There is a 2σ hint for θ13 being non-zero
Fogli et al ‘08
Fogli et al ‘09
The 2009 estimate includes the MINOS results which show a 1.5σ
excess of events in the electron appearance channel
04/02/2010
Steve King, University of Warwick, Coventry
Neutrino mass squared splittings and angles
Normal
Inverted
Absolute neutrino mass scale?
θ12 = 34.5° ± 1.4°
Why are neutrino
masses so small ?
04/02/2010
θ 23 = 43 ± 4
°
°
θ13 ≤ 10°
Steve King, University of Warwick, Coventry
Two of these angles are
pretty large – why?
Tri-bimaximal mixing matrix UTB
Harrison, Perkins, Scott
TB angles
c.f. data
θ12 = 35 ,
θ 23 = 45 ,
θ13 = 0 .
θ12 = 34.5° ± 1.4 , θ 23 = 43.1° ± 4o , θ13 = 8° ± 2o
Current data is consistent with TB mixing
(ignoring the 2σ hint for θ13 )
04/02/2010
Steve King, University of Warwick, Coventry
Useful to parametrize the PMNS mixing matrix in terms of
deviations from TBM
SFK
‘07
r = reactor
s = solar
a = atmospheric
e.g. r≠ 0, s=0, a=0 gives Tri-bimaximal-reactor (TBR) mixing
SFK
‘09
TBR not as simple as TB but is required if θ13≠ 0
04/02/2010
Steve King, University of Warwick, Coventry
Leptonic CP violation is unknown
r
Scaled triangle
δ
Neither r nor δ is
measured – UT could
be a straight line!
1
04/02/2010
Steve King, University of Warwick, Coventry
Neutrinos can have Dirac and/or Majorana Mass
CP conjugate
Majorana masses
mLLν ν
c
L L
M ν ν
c
RR R R
mLRν Lν R
Dirac mass
04/02/2010
Steve King, University of Warwick, Coventry
Violates L
Violates Le , Lμ , Lτ
Neutrino=antineutrino
Conserves L
Violates Le , Lμ , Lτ
Neutrino ≠antineutrino
In general a mass term can be thought of as an
interaction between left and right-handed chiral fields
−
−
L
R
e
e
− −
e L R
me e
Left-handed neutrinos νL can form masses with either righthanded neutrinos νR or with their own CP conjugates νcL
νL
ν
mLRν Lν R
Majorana
04/02/2010
νR
Dirac
ν
m νν
c
LL L L
ν
Steve King, University of Warwick, Coventry
c
L
Right-handed neutrinos νR can also form masses with their
own CP conjugates ν c
R
νR
ν
c
R
M ν ν
c
RR R R
Majorana
In principle there is nothing to prevent the right-handed
Majorana mass MRR from being arbitrarily large since νR is
a gauge singlet e.g. MRR=MGUT
On the other hand it is possible that MRR =0 which could
be enforced by lepton number L conservation
04/02/2010
Steve King, University of Warwick, Coventry
Absolute ν mass scale and the nature of ν mass
Neutrinoless
double beta
decay
Tritium beta
decay
Present Mainz < 2.2 eV
KATRIN
| mee |=
∑U
Majorana (no signal if Dirac)
2
ei
mi
i
| mν e | = Σi | Uei |2 mi
~0.35eV
U e1 = c12 c13eiα1 / 2 , U e 2 = s12 c13eiα 2 / 2 , U e3 = s13e − iδ
04/02/2010
Steve King, University of Warwick, Coventry
KlapdorMajorana
GERDA
76
Ge
76 Ge
76
Ge
~0.4 eV (signal)
~0.05 eV
~0.1 eV (phase II)
~0.01 eV (phase III )
Non- 76 Ge:
CUORE, NEMO3
SuperNEMO
Æ50meV=0.05 eV
COBRA Cd116 Æ??
EXO Xe136 1 ton 5y 50-70meV=0.05-0.07eV
SNO+ Nd150 start 2011? Æ50meV=0.05 eV
The Future
I 100-500meV SNemo CUORE GERDA EXO SNO+
II 15-50meV 1 ton 10y?
III 2-5meV 100 tons 20y??
Cosmology vs Neutrinoless DBD
0ν DBD
Approx. degeneracy
is TESTABLE
Inverted hierarchy is
TESTABLE
future
cosmo
Normal hierarchy is
NOT TESTABLE
from: F. Feruglio, A. Strumia, F. Vissani ('02)
04/02/2010
Steve King, University of Warwick, Coventry
Part III.
Flavour in the
Standard Model
04/02/2010
Steve King, University of Warwick, Coventry
Yukawa matrices
Hψ Y ψ
i
L ij
j
R
Yij → Y , Y , Y , Y
U
ij
e. g . YijE (ν e
D
ij
E
ij
+
⎛
⎞ j
H
i
e ) L ⎜ 0 ⎟ eR
⎝H ⎠
N
ij
ψ
ψ
Quark mixing matrix VCKM
⎛ mu
V ULYLRUV UR †v = ⎜⎜ 0
⎜0
⎝
0
mc
0
0⎞
0 ⎟⎟
mt ⎟⎠
⎛ md
V DLYLRDV DR †v = ⎜⎜ 0
⎜ 0
⎝
VCKM = V ULV DL †
Defined as
0
ms
0
0⎞
0 ⎟⎟
mb ⎟⎠
5 phases removed
Lepton mixing matrix UPMNS
Light neutrino Majorana mass matrix
⎛ me
V ELYLREV ER †v = ⎜⎜ 0
⎜0
⎝
0
mμ
0
Defined as
0⎞
0 ⎟⎟
mτ ⎟⎠
ν
V ν L mLL
V ν LT
⎛ m1 0 0 ⎞
⎜
⎟
= ⎜ 0 m2 0 ⎟
⎜0 0 m ⎟
3⎠
⎝
UPMNS = V ELV ν L † 3 phases removed
Recall the origin of the electron mass in
the SM are the Yukawa couplings:
y (ν e
e
⎛H+⎞
e ) L ⎜ 0 ⎟ eR
⎝H ⎠
y (ν e
e
⎛< H + >⎞ ⎛0⎞
⎜
⎟=⎜ ⎟
0
⎝< H >⎠ ⎝v⎠
⎛ 0⎞
e ) L ⎜ ⎟ eR = y e v eL eR
⎝v⎠
me
Yukawa coupling ye must be small since <H0>=v=175 GeV
me = ye H 0 ≈ 0.5 MeV ⇔ ye ≈ 3.10−6
Unsatisfactory
Introduce right-handed neutrino νeR with zero Majorana mass
yν LH cν eR = yν H 0 ν eLν eR
then Yukawa coupling generates a Dirac neutrino mass
mνLR = yν H 0 ≈ 0.2 eV ⇔ yν ≈ 10−12
Even more unsatisfactory
Part IV.
The Origin of
Neutrino Mass
04/02/2010
Steve King, University of Warwick, Coventry
Three important features of the SM
1. There are no right-handed neutrinos
νR
2. There are only Higgs doublets of SU(2)L
3. There are only renormalizable terms
In the Standard Model neutrinos are massless, with νe , νμ , ντ
distinguished by separate lepton numbers Le, Lμ, Lτ
Neutrinos and anti-neutrinos are distinguished by the total
conserved lepton number L=Le+Lμ+Lτ
To generate neutrino mass we must relax 1 and/or 2 and/or 3
04/02/2010
Steve King, University of Warwick, Coventry
Neutrino Mass Models Road Map
Dirac
Dirac or Majorana?
Extra dims?
Majorana
See-saw mechanisms, Higgs Triplets, Loops, RPV, …
Very precise TB?
Yes
Family symmetry?
Yes
Degenerate?
No
Anarchy, see-saw, etc…
No
Yes
Alternatives?
Type II see-saw?
No
Type I see-saw?
04/02/2010
GUTs and/or Strings?
Steve King, University of Warwick,
Coventry
Origin of Majorana Neutrino Mass
Renormalisable
ΔL =2 operator
λν LLΔ
Non-renormalisable
ΔL =2 operator
λν
M
where Δ is light Higgs triplet with
VEV < 8GeV from ρ parameter
LLHH =
λν
M
H
0 2
ν ν
c
eL eL Weinberg
This is nice because it gives naturally small Majorana neutrino
masses mLL∼ <H0>2/M where M is some high energy scale
The high mass scale can be associated with some heavy
particle of mass M being exchanged (can be singlet or triplet)
H
L
04/02/2010
M
H
H
H
M
L
L
e.g. see-saw mechanism
L
Steve King, University of Warwick, Coventry
The See-Saw Mechanism
Light neutrinos
Heavy particles
04/02/2010
Steve King, University of Warwick,
Coventry
The see-saw mechanism
Type I see-saw mechanism
P. Minkowski (1977), Gell-Mann, Glashow,
Mohapatra, Ramond, Senjanovic, Slanski,
Yanagida (1979/1980)
Type II see-saw mechanism (SUSY)
Lazarides, Magg, Mohapatra, Senjanovic,
Shafi, Wetterich (1981)
λΔ
νR
νL
Heavy
triplet
νL
νL
M νν
c
RR R R
−1
RR
m ≈ −mLR M m
I
LL
Type I
T
LR
YΔ
νL
2
u
v
m ≈ λΔYΔ
MΔ
II
LL
Type II
The See-Saw Matrix
Dirac matrix
Type II contribution
(
ν L ν Rc
)
II
⎛ mLL
⎜ T
⎝ mLR
mLR ⎞ ⎛ν Lc ⎞
⎟⎜ ⎟
M RR ⎠ ⎝ν R ⎠
Heavy Majorana matrix
Diagonalise to give effective mass
Light Majorana matrix
→m ν ν
c
LL L L
II
−1 T
mνLL ≈ mLL
− mLR M RR
mLR
2
mνLL ∼ mLR
/ M RR suggests new high energy mass scale(s) Æ radiative corrections
Part V.
Flavour Models
04/02/2010
Steve King, University of Warwick, Coventry
Hierarchical Symmetric Textures
Consider the following ansatz for the upper 2x2 block of a hierarchical Yd
⎡⎣Y ⎤⎦
d
ij 1− 2
⎛ 0
=⎜
⎝ y12
y12 ⎞
y22 ⎟⎠
| Vus |≈
md
≈λ
ms
Gatto et al
successful
prediction
This motivates having a symmetric down quark Yukawa matrix with a
1-1 “texture zero ” and a hierarchical form
⎛0
⎜ 3
d
Y ∼ ⎜λ
⎜λ3
⎝
λ3 λ3 ⎞
2
2⎟
λ λ ⎟ yb
λ 2 1 ⎟⎠
D
( mLR
) 23
| Vcb |≈
≈ λ2
mb
λ ≈ 0.2 is the Wolfenstein Parameter
D
( mLR
)13
| Vub |≈
≈ λ3
mb
successful
predictions
G.Ross et al
Up quarks are more hierarchical than down quarks
This suggests different expansion parameters for up and down
YLRD
⎛ 0 ε d3
⎜
∼ ⎜ ε d3 ε d2
⎜ ε d3 ε d2
⎝
ε d3 ⎞
⎟
ε d2 ⎟ yb
1 ⎟⎠
ε d ∼ 0.15
md : ms : mb = ε d4 : ε d2 :1
U
YLR
⎛ 0 ε u3
⎜
∼ ⎜ ε u3 ε u2
⎜ ε u3 ε u2
⎝
ε u3 ⎞
⎟
ε u2 ⎟ yt
1 ⎟⎠
ε u ∼ 0.05
mu : mc : mt = ε u4 : ε u2 :1
Charged leptons are well described by similar matrix to the downs but with
a numerical factor of about 3 in the 2-2 entry (Georgi-Jarlskog)
E
LR
Y
⎛0
⎜
∼ ⎜ ε d3
⎜ ε d3
⎝
ε d3 ε d3 ⎞
⎟
3ε d2 ε d2 ⎟ yb
ε d2 1 ⎟⎠
md
me : mμ : mτ =
: 3ms : mb
ε d ∼ 0.15
3
md
mb
RGE → me ≈
, mμ ≈ ms , mτ ≈
9
3
at MU
at mb
N.B. Electron mass is governed by an expansion parameter εd ∼ 0.15 which is not
unnaturally small – providing we can generate these textures from a theory
Textures from U(1) Family Symmetry
φ ≠0
For D-flatness we use a pair of flavons with opposite U(1) charges Q (φ ) = −Q (φ )
Consider a U(1) family symmetry spontaneously broken by a flavon vev
Example: U(1) charges as Q (ψ3 )=0, Q (ψ2 )=1, Q (ψ1 )=3, Q(H)=0, Q(φ )=-1,Q(φ)=1
⎛ 0 0 0⎞
Then at tree level the only allowed Yukawa coupling is H ψ3 ψ3 → Y = ⎜ 0 0 0 ⎟
⎜
⎟
⎜0 0 1⎟
⎝
⎠
The other Yukawa couplings are generated from higher order operators which
respect U(1) family symmetry due to flavon φ insertions:
−1 + 0 + 1 + 0 = 0
φ
⎛ φ ⎞
2
⎛ φ ⎞
3
⎛ φ ⎞
4
⎛ φ ⎞
6
Hψ 2ψ 3 + ⎜ ⎟ Hψ 2ψ 2 + ⎜ ⎟ Hψ 1ψ 3 + ⎜ ⎟ Hψ 1ψ 2 + ⎜ ⎟ Hψ 1ψ 1
M
⎝M ⎠
⎝M ⎠
⎝M ⎠
⎝M ⎠
When the flavon gets its VEV it generates
small effective Yukawa couplings in terms
of the expansion parameter
ε=
φ
M
⎛ε 6 ε 4 ε 3 ⎞
⎜
⎟
→ Y = ⎜ε 4 ε 2 ε ⎟
⎜ε 3 ε 1 ⎟
⎝
⎠
Not quite of
desired form
Æ non-Abelian
Froggatt-Nielsen Mechanism
What is the origin of the higher order operators?
Froggat and Nielsen took their inspiration from the see-saw mechanism
H
H
2
M
Mν R
νL
νR
νR νL
φ
H
→
M
Mχ
ψ2
H
→
ν Lν L
Mν R
χ
χ
ψ3
φ
Mχ
Hψ 2ψ 3
Where χ are heavy fermion messengers
c.f. heavy RH neutrinos
There may be Higgs messengers or fermion messengers
H0
φ
H1
H
−1
ψ2
φ −1
−1
H0
Mχ
MH
ψ2
ψ3
χ0
χ0
ψ3
Fermion messengers may be SU(2)L doublets or singlets
φ
−1
H
0
H
Mχ
M χQ
Q2
04/02/2010
χ Q0
χ Q0
φ −1
0
U c3
Q2
Uc
χ U− 1 χ U1
c
Steve King, University of Warwick,
Coventry
c
U c3
SFK, Ross,Varzielas
Textures from SU(3) Family Symmetry
In SU(3) with ψi=3 and H=1 all tree-level Yukawa couplings Hψi ψj are forbidden.
SU (3)
Ytree
− level
In SU(3) with flavons
⎛ 0 0 0⎞
= ⎜⎜ 0 0 0 ⎟⎟
⎜ 0 0 0⎟
⎝
⎠
φ i = 3 the lowest order Yukawa operators allowed are:
1 i j
φ φ Hψ iψ j
2
M
For example suppose we consider a flavon
generates a (3,3) Yukawa coupling
φ3i with VEV φ3i = (0,0,1)u3 then this
⎛ 0 0 0⎞ 2
1 i j
u3
⎜
⎟
φ φ Hψ iψ j → ⎜ 0 0 0 ⎟ 2
2 3 3
M
⎜0 0 1⎟ M
⎝
⎠
Note that we label the flavon φ3 with a subscript 3 which denotes the
direction of its VEV in the i=3 direction.
i
Next suppose we consider a flavon φ23 with VEV φ23i = (0,1,1)u2 then this
generates (2,3) block Yukawa couplings
i
⎛ 0 0 0⎞ 2
1
⎜ 0 1 1 ⎟ u2
i
j
H
φ
φ
ψ
ψ
→
i j
23 23
⎜
⎟ M2
M2
⎜0 1 1⎟
⎝
⎠
i
i
To complete the model we use a flavon φ123 with VEV φ123 = (1,1,1)u1 then this
generates Yukawa couplings in the first row and column
⎛0 1 1⎞
1
⎜ 1 − − ⎟ u1u2
i
j
H
φ
φ
ψ
ψ
→
i j
123 23
⎜
⎟ M2
M2
⎜1 − −⎟
⎝
⎠
u32
Taking 1 ∼
M2
u2 2
ε ∼ 2
M
2
⎛ 0 0 0 ⎞ φ3 ≠ 0
⎜ 0 0 0⎟
⎜
⎟
⎜ 0 0 0⎟
⎝
⎠
ε3 ∼
⎛ 0 0 0⎞
⎜ 0 0 0⎟
⎜
⎟
⎜0 0 1⎟
⎝
⎠
u1u2
we generate desired structures
M2
φ23 ≠ 0 ⎛ 0
0
⎜0 ε
⎜
⎜0 ε
⎝
2
2
0 ⎞ φ123 ≠ 0
ε 2 ⎟⎟
1 ⎟⎠
⎛ 0 ε3 ε3 ⎞
⎜ 3
2
2⎟
ε
ε
ε
⎜
⎟
3
2
⎜ε ε
⎟
1
⎝
⎠
Froggatt-Nielsen diagrams
Right-handed fermion messengers dominate with Mu∼ 3 Md
⎛0 0
1
⎜
i
j
2
→
φ
φ
H
ψ
ψ
0
ε
23 23
i j
u ,d
⎜
M u2,d
⎜0 ε2
u ,d
⎝
0 ⎞
2 ⎟
ε u ,d ⎟
ε u2,d ⎟⎠
εd = <φ23>/Md ∼ 0.15 and εu = <φ23>/Mu ∼ 0.05
Unsatisfactory features:
1. Suggests yb>yt !
2. First row/column is only quadratic in the messenger mass.
To improve model we introduce sextet and singlet flavons
SFK,Luhn’09
Flavon sextets for the third family
Idea is to use flavon sextets χ = 6 and Higgs messengers to
generate the third family Yukawa couplings
χ 33 = V
1
Hψ i χ ijψ j
MH
Third family
Yukawas
04/02/2010
Steve King, University of Warwick,
Coventry
⎛ 0 0 0⎞
⎜ 0 0 0⎟ V
⎜
⎟M
⎜0 0 1⎟ H
⎝
⎠
yt ∼ yb ∼ V / M H
SFK,Luhn’09
Flavon singlet for first row/column
Flavon singlet ξ = 1 leads to first row and column Yukawa
couplings involving a cubic messenger mass
⎛ 0
1
⎜ 3
i
j
ξφ
φ
H
ψ
ψ
ε u ,d
→
123 23
i j
3
⎜
M u ,d
⎜ ε u3,d
⎝
ε u3,d ε u3,d ⎞
−
−
⎟
− ⎟
− ⎠⎟
So finally we arrive at desired quark textures (with yt∼ yb )
YLRD
⎛ 0 ε d3
⎜
∼ ⎜ ε d3 ε d2
⎜ ε d3 ε d2
⎝
04/02/2010
ε d3 ⎞
⎟
ε d2 ⎟ yb
1 ⎟⎠
ε d ∼ 0.15
U
YLR
⎛ 0 ε u3
⎜
∼ ⎜ ε u3 ε u2
⎜ ε u3 ε u2
⎝
Steve King, University of Warwick,
Coventry
ε u3 ⎞
⎟
ε u2 ⎟ yt
1 ⎟⎠
ε u ∼ 0.05
Part VI.
Discrete Family
Symmetry
04/02/2010
Steve King, University of Warwick, Coventry
Discrete neutrino flavour symmetry
⎛ 1
Consider the TB
ν
= U TB ⎜⎜ 0 m2
M TB
Neutrino Mass Matrix
⎜
m
0
⎝0
0
0⎞
T
0 ⎟⎟ U TB
m3 ⎟⎠
ν
M TB
=
TB Neutrino Mass
Matrix is invariant
under a discrete
Z2× Z2 group
generated by S,U
04/02/2010
ν
ν
M TB
= S M TB
ST
ν
ν
M TB
= U M TB
UT
Steve King, University of Warwick, Coventry
Lam
S4 family symmetry
In this basis the charged lepton matrix is invariant under a diagonal
phase symmetry T
⎛ me
M E = ⎜⎜ 0
⎜0
⎝
0
mμ
0
0⎞
0 ⎟⎟ = TM ET
mτ ⎟⎠
†
⎛ 1 0 0⎞
⎜
⎟
T = ⎜ 0 ω2 0 ⎟
⎜0 0 ω⎟
⎝
⎠
ω = e 2π i / 3
S,T,U Æ generate the discrete group S4
Suggests using a discrete family symmetry S4 broken by three types of
flavons φS , φT, φU which each preserve a particular generator
Charged leptons preserve T
Neutrinos preserve S,U
04/02/2010
Steve King, University of Warwick, Coventry
SFK,
Luhn
Indirect models
Alternatively it is possible to realise the neutrino flavour symmetry
indirectly as an accidental symmetry
Introduce three triplet flavons φ1 , φ2 , φ3 with VEVs along columns of UTB
These flavons
break S,U
The following Majorana Lagrangian preserves S,U accidentally
04/02/2010
Steve King, University of Warwick, Coventry
GFami l y
SU(3)
Δ27
P S L2 ( 7 )
Δ 96
Z2 × Z 7
S4
A4
Discrete family symmetry
preferred by:
- Vacuum alignment
- String theory
04/02/2010
Steve King, University of Warwick, Coventry
SO(3)
Part VII. GUT Models
Strong
Weak
Electromagnetic
04/02/2010
Steve King, University of Warwick, Coventry
GUTs and Family Symmetry
™Gauge coupling unification suggests GUTs
™TB mixing suggests a discrete family symmetry
×10−3
ν1
< ×10
−12
Family
symmetry
04/02/2010
×10−3
ν2
×10
c
×10−1
s
μ
×10−4
×1
u
d
e
t
b
×10−2
×10
−2
−12
ν3
τ×10
×10−1
−1
×10−11
Steve King, University of Warwick, Coventry
GUT
symmetry
GGUT
E6
SU (5) × U (1)
SO(10)
SU (3)C × SU (3) L × SU (3) R
SU (4) PS × SU (2) L × SU (2) R
SU (5)
SU (3)C × SU (2) L × SU (2) R × U (1) B − L
SU (3)C × SU (2) L × U (1)Y
04/02/2010
Steve King, University of Warwick, Coventry
GUT relations and sum rule
Georgi-Jarlskog
.
.
.
TB + charged lepton
.
corrections + GUTs
. : θ13
leads to predictions
.
04/02/2010
e
θ12 ≈
≈
θC
3 2
≈ 3° ,
Bjorken; Ferrandis, Pakvasa; SFK
d
.12
θ
3
θ12 ≈ 35.3o + θ13 cosδ
SFK, Antusch, Masina,
Malinsky, Boudjemaa
Steve King, University of Warwick, Coventry
≈
θC
3
Sum Rule
θ12 ≈ 35.3 + θ13 cosδ
o
Testing the sum rule
.
Bands show 3σ error
for optimized
neutrino factory
determination of
θ13cos δ
θ12=34.5o± 1.4o
.
(current value)
Antusch,
Huber,
SFK,
Schwetz
04/02/2010
Steve King, University of Warwick, Coventry
Conclusions
„
Neutrino mass and mixing is first new physics BSM
and adds impetus to solving the flavour problem
„
Many possible origins of neutrino mass, but focus on
ideas which may lead to a theory of flavour: see-saw
mechanism and family symmetry broken by flavons
„
If TB mixing is accurately realised this may imply
discrete family symmetry
„
GUTs × discrete family symmetry with see-saw is very
attractive framework for TB mixing
04/02/2010
Steve King, University of Warwick, Coventry
SFK,Luhn’09
PSL2(7) x SO(10) Model
Attractive features of PSL2(7):
- The smallest simple finite group containing complex triplets and sextets
- Contains S4 as subgroup (i.e. contains the generators S,T,U)
- Contains sextet reps Æ allows flavon sextets χ = 6 used both for third
family Yukawas and for TB mixing where <χTB> preserve S,U
- Flavon triplets φ23, φ123 and flavon singlet ξ as before
Yukawa Sector (as before)
Majorana Sector (new)
c
Δ126
∼ M GUT
M RR ∼
Type II see-saw is dominant
Very heavy Æ type I
see-saw is subdominant
Type II neutrino phenomenology
mee
mee
mmin
mmin
∑m
∑m
i
i
mmin
mmin
Indirect type I see-saw models
Consider
Example of Form dominance Æ TB mixing independently of masses
Constrained sequential dominance corresponds to m1Æ0
04/02/2010
Steve King, University of Warwick, Coventry
SFK