1
.
Flavour Permutation Symmetry and Fermion Mixing
Paul Harrison∗
University of Warwick
Seminar at University of Birmingham,
30th April, 2008
∗
With Dan Roythorne and Bill Scott, PLB 657 (2007) 210, arXiv:0709.1439
Paul Harrison
University of Warwick
Birmingham, 30th April 2008
2
.
Outline of Talk
• Introduction
• What does the SM tell us about flavour?
• What do the Data tell us about flavour?
• Origin of Masses and Mixings in SM
• Plaquette Invariance and Flavour Permutation Symmetry
• Flavour Symmetric Mixing Observables
• Applications
• Conclusions
Paul Harrison
University of Warwick
Birmingham, 30th April 2008
3
Thinking about Flavour
Gauge sector of SM is rather well-understood. Has 3 free observable parameters.
Paul Harrison
University of Warwick
Birmingham, 30th April 2008
3
Thinking about Flavour
Gauge sector of SM is rather well-understood. Has 3 free observable parameters.
The flavour sector is very poorly understood. Has 20 (or 22) low energy observable
(free) parameters. Accounts for huge phenomenological richness - hierarchies of:
• masses
• mixing angles
• lifetimes
• flavour oscillations
• CP violation
• etc...
Spent many years measuring some of these parameters.
Paul Harrison
University of Warwick
Birmingham, 30th April 2008
3
Thinking about Flavour
Gauge sector of SM is rather well-understood. Has 3 free observable parameters.
The flavour sector is very poorly understood. Has 20 (or 22) low energy observable
(free) parameters. Accounts for huge phenomenological richness - hierarchies of:
• masses
• mixing angles
• lifetimes
• flavour oscillations
• CP violation
• etc...
Spent many years measuring some of these parameters.
But (frustratingly) there still exists no convincing, accepted theory of flavour (and I’m
not going to propose one today!).
Paul Harrison
University of Warwick
Birmingham, 30th April 2008
4
Open Questions
What we don’t know (yet) includes:
• Why 3 families?
Paul Harrison
University of Warwick
Birmingham, 30th April 2008
4
Open Questions
What we don’t know (yet) includes:
• Why 3 families?
• Why the flavour parameters take the values that they do?
Paul Harrison
University of Warwick
Birmingham, 30th April 2008
4
Open Questions
What we don’t know (yet) includes:
• Why 3 families?
• Why the flavour parameters take the values that they do?
• Whether neutrinos are Dirac or Majorana (a big problem if you want to construct a
theory of quark and lepton flavour).
Paul Harrison
University of Warwick
Birmingham, 30th April 2008
4
Open Questions
What we don’t know (yet) includes:
• Why 3 families?
• Why the flavour parameters take the values that they do?
• Whether neutrinos are Dirac or Majorana (a big problem if you want to construct a
theory of quark and lepton flavour).
• Whether the whole theory is embedded in some BSM theory - SUSY, strings, extra
dimensions...We (most of us) assume it is so embedded, but we have no idea in
what.
Many people make assumptions about the above and try to make viable models.
Paul Harrison
University of Warwick
Birmingham, 30th April 2008
4
Open Questions
What we don’t know (yet) includes:
• Why 3 families?
• Why the flavour parameters take the values that they do?
• Whether neutrinos are Dirac or Majorana (a big problem if you want to construct a
theory of quark and lepton flavour).
• Whether the whole theory is embedded in some BSM theory - SUSY, strings, extra
dimensions...We (most of us) assume it is so embedded, but we have no idea in
what.
Many people make assumptions about the above and try to make viable models.
This is not our approach.
Paul Harrison
University of Warwick
Birmingham, 30th April 2008
5
Our Approach
• Wanted to be more general.
Paul Harrison
University of Warwick
Birmingham, 30th April 2008
5
Our Approach
• Wanted to be more general.
• Notwithstanding the open questions, try to see what we can say about flavour
“BSM”, taking only the data and the SM as our guide
Paul Harrison
University of Warwick
Birmingham, 30th April 2008
5
Our Approach
• Wanted to be more general.
• Notwithstanding the open questions, try to see what we can say about flavour
“BSM”, taking only the data and the SM as our guide
• Make as few assumptions as possible (no prejudice, least likely to be wrong!).
Paul Harrison
University of Warwick
Birmingham, 30th April 2008
5
Our Approach
• Wanted to be more general.
• Notwithstanding the open questions, try to see what we can say about flavour
“BSM”, taking only the data and the SM as our guide
• Make as few assumptions as possible (no prejudice, least likely to be wrong!).
• See what the data suggest - an experimentalist’s approach.
Paul Harrison
University of Warwick
Birmingham, 30th April 2008
5
Our Approach
• Wanted to be more general.
• Notwithstanding the open questions, try to see what we can say about flavour
“BSM”, taking only the data and the SM as our guide
• Make as few assumptions as possible (no prejudice, least likely to be wrong!).
• See what the data suggest - an experimentalist’s approach.
• Very much work in progress.
Paul Harrison
University of Warwick
Birmingham, 30th April 2008
6
Start with the Standard Model
What does SM tell us about flavour?
Paul Harrison
University of Warwick
Birmingham, 30th April 2008
6
Start with the Standard Model
What does SM tell us about flavour?
Nothing!
Paul Harrison
University of Warwick
Birmingham, 30th April 2008
6
Start with the Standard Model
What does SM tell us about flavour?
Nothing!
Well, almost.
Paul Harrison
University of Warwick
Birmingham, 30th April 2008
6
Start with the Standard Model
What does SM tell us about flavour?
Nothing!
Well, almost.
 
 
u
νe



.
Places two quarks in a weak isodoublet ψq =
and two leptons: ψℓ =
d
e
Paul Harrison
University of Warwick
Birmingham, 30th April 2008
6
Start with the Standard Model
What does SM tell us about flavour?
Nothing!
Well, almost.
 
 
u
νe



.
Places two quarks in a weak isodoublet ψq =
and two leptons: ψℓ =
d
e
They participate in the standard gauge-invariant Electroweak interaction, via covariant
derivatives etc. (will not review).
Paul Harrison
University of Warwick
Birmingham, 30th April 2008
6
Start with the Standard Model
What does SM tell us about flavour?
Nothing!
Well, almost.
 
 
u
νe



.
Places two quarks in a weak isodoublet ψq =
and two leptons: ψℓ =
d
e
They participate in the standard gauge-invariant Electroweak interaction, via covariant
derivatives etc. (will not review).
Distinguished by their charges...
Paul Harrison
University of Warwick
Birmingham, 30th April 2008
7
...and by their Masses
LY uk ∼ cd ψ qL · φ · dR + cu ψ qL · φ̃ · uR
+ ce ψ ℓL · φ · eR + cν ψ ℓL · φ̃ · νeR (+any Majorana mass terms...)
with “Yukawa couplings” cu , cd , ce , cν free parameters (to be determined by
experiment).
Paul Harrison
University of Warwick
Birmingham, 30th April 2008
7
...and by their Masses
LY uk ∼ cd ψ qL · φ · dR + cu ψ qL · φ̃ · uR
+ ce ψ ℓL · φ · eR + cν ψ ℓL · φ̃ · νeR (+any Majorana mass terms...)
with “Yukawa couplings” cu , cd , ce , cν free parameters (to be determined by
experiment).
After SSB,
OR, if Majorana ν, then
Paul Harrison
mi = ci .(vev).
mν =
c2ν (vev)2
.
MM aj
University of Warwick
Birmingham, 30th April 2008
7
...and by their Masses
LY uk ∼ cd ψ qL · φ · dR + cu ψ qL · φ̃ · uR
+ ce ψ ℓL · φ · eR + cν ψ ℓL · φ̃ · νeR (+any Majorana mass terms...)
with “Yukawa couplings” cu , cd , ce , cν free parameters (to be determined by
experiment).
After SSB,
OR, if Majorana ν, then
mi = ci .(vev).
mν =
c2ν (vev)2
.
MM aj
SM accommodates multiple copies of this - families/generations.
Then Yukawas, ci , become arbitrary matrices: Y 2 , Y− 1 , Yℓ , Yν .
3
Paul Harrison
University of Warwick
3
Birmingham, 30th April 2008
7
...and by their Masses
LY uk ∼ cd ψ qL · φ · dR + cu ψ qL · φ̃ · uR
+ ce ψ ℓL · φ · eR + cν ψ ℓL · φ̃ · νeR (+any Majorana mass terms...)
with “Yukawa couplings” cu , cd , ce , cν free parameters (to be determined by
experiment).
After SSB,
OR, if Majorana ν, then
mi = ci .(vev).
mν =
c2ν (vev)2
.
MM aj
SM accommodates multiple copies of this - families/generations.
Then Yukawas, ci , become arbitrary matrices: Y 2 , Y− 1 , Yℓ , Yν .
3
3
And that’s all!
Paul Harrison
University of Warwick
Birmingham, 30th April 2008
8
And the Data? 3 Families =⇒ 2 Quark Mass Spectra
Charge = − 13
103
102
2
3
t ~ 174 GeV
10
1
Mass (GeV)
Mass (GeV)
Charge = + 32
c ~ 1.3 GeV
10-1
10-2
10
-1
3
b ~ 4.5 GeV
1
s ~ 0.175 GeV
10-1
10-2
d ~ 0.010 GeV
u ~ 0.005 GeV
10-3
10-3
Apparently not a “typical” random, arbitrary set of parameters!
Paul Harrison
University of Warwick
Birmingham, 30th April 2008
9
The Data - 2 Lepton Mass Spectra
Charge = −1
Charge = 0
10
1
10-1
τ ~ 1.777 GeV
µ ~ 0.105 GeV
∆ Mass2 (eV2)
Mass (GeV)
-1
1
ν
10-1
|m2 - m21| ~ 2.5 × 10-3 eV2
3
10-2
10-2
|m2 - m21| ~ 8 × 10 eV2
2
-5
-3
10
e ~ 0.0005 GeV
10-3
10-4
Similarly not a “typical” random, arbitrary set of parameters
Paul Harrison
University of Warwick
Birmingham, 30th April 2008
10
The Data - CKM Quark Mixing Matrix
• CKM matrix parameterises couplings of quark mass eigenstates to W .
• Shows tantalising “Wolfenstein” structure:
d
u

O(1)


|V | ∼ c  O(λ)

t
O(λ3 )
s
O(λ)
O(1)
O(λ2 )
b
O(λ3 )



O(λ2 ) 

O(1)
+
W
qi
Vi α
qα
• Elements not predictable in the SM - profoundly unsatisfactory.
.
Paul Harrison
University of Warwick
Birmingham, 30th April 2008
11
Summary of Neutrino Oscillation Data
1.4
Solar
Reactor
1.2
Nobs/Nexp
1.0
0.8
ILL
Savannah River
Bugey
Rovno
Goesgen
Krasnoyarsk
Palo Verde
Chooz
0.6
0.4
0.2
KamLAND
0.0
101
ν1
e

∼


(|U |2 ) = µ  ∼

∼
τ
Paul Harrison
2
3
1
6
1
6
ν2
ν3
0.31 ± 0.04
∼
∼
1
3
1
3
< 0.013
102
103
104
Distance to Reactor (m)
105
Atmospheric



0.50 ± 0.11

∼ 12
University of Warwick
Birmingham, 30th April 2008
12
TriBimaximal Lepton Mixing
So, in leading approximation:
ν1
e



|U | ∼ µ 

τ
√
2/ 6
√
1/ 6
√
1/ 6
ν2
√
1/ 3
√
1/ 3
√
1/ 3
ν3
0
√
1/ 2
√
1/ 2





PFH, D. Perkins and W.G. Scott, PLB 458 (1999) 79 and PLB 530 (2002) 167.
Paul Harrison
University of Warwick
Birmingham, 30th April 2008
12
TriBimaximal Lepton Mixing
So, in leading approximation:
ν1
e



|U | ∼ µ 

τ
√
2/ 6
√
1/ 6
√
1/ 6
ν2
√
1/ 3
√
1/ 3
√
1/ 3
ν3
0
√
1/ 2
√
1/ 2





PFH, D. Perkins and W.G. Scott, PLB 458 (1999) 79 and PLB 530 (2002) 167.
Or (more realistically):

√2 (1
6
− ǫ)
√1 (1
3
+ 2ǫ)


∗
√1 (1 − ǫ − µ)
U ≃ − √1 (1 + 2ǫ − µ + √3 Ue3
)
2
3
 6
√1 (1 + 2ǫ + µ − √3 U ∗ )
√1 (1 − ǫ + µ)
−
6
2 e3
3
Ue3



√1 (1 + µ − √1 Ue3 ) .
2
2

√1 (1 − µ + √1 Ue3 )
2
2
with
Paul Harrison
−0.032 < ǫ < 0.037;
−0.18 < µ < 0.21;
University of Warwick
|Ue3 | < 0.11.
Birmingham, 30th April 2008
13
TBM Defined by 3 Phenomenological Symmetries
• CP :
If Im(Ue3 ) = 0, then U is real and CP is conserved.
Paul Harrison
University of Warwick
Birmingham, 30th April 2008
13
TBM Defined by 3 Phenomenological Symmetries
• CP :
If Im(Ue3 ) = 0, then U is real and CP is conserved.
• Democracy:
If ǫ = µ = 0, then middle column is ν2 =
√1 (1, 1, 1)
3
Explored in J.D. Bjorken, PFH, and W.G. Scott PRD 74 (2006) 073012.
Paul Harrison
University of Warwick
Birmingham, 30th April 2008
13
TBM Defined by 3 Phenomenological Symmetries
• CP :
If Im(Ue3 ) = 0, then U is real and CP is conserved.
• Democracy:
If ǫ = µ = 0, then middle column is ν2 =
√1 (1, 1, 1)
3
Explored in J.D. Bjorken, PFH, and W.G. Scott PRD 74 (2006) 073012.
• µ − τ Reflection Symmetry:
If Re(Ue3 ) = 0 = µ, then |Uµi | = |Uτ i | and simulataneous µ − τ swap and CP
transf. leaves observables invariant (PFH and W.G. Scott PLB 547 (2002) 219).
Paul Harrison
University of Warwick
Birmingham, 30th April 2008
13
TBM Defined by 3 Phenomenological Symmetries
• CP :
If Im(Ue3 ) = 0, then U is real and CP is conserved.
• Democracy:
If ǫ = µ = 0, then middle column is ν2 =
√1 (1, 1, 1)
3
Explored in J.D. Bjorken, PFH, and W.G. Scott PRD 74 (2006) 073012.
• µ − τ Reflection Symmetry:
If Re(Ue3 ) = 0 = µ, then |Uµi | = |Uτ i | and simulataneous µ − τ swap and CP
transf. leaves observables invariant (PFH and W.G. Scott PLB 547 (2002) 219).
• CP and Democracy and µ − τ Reflection Symmetry
⇒ Ue3 = 0 = ǫ = µ ⇒ TBM mixing.
Paul Harrison
University of Warwick
Birmingham, 30th April 2008
14
Discussion of TBM
TBM form is redolent of other symmetries:
• Taking νe , νµ , ντ to define the orientation of a cube, ν2 lies along its
body diagonal
• Same coefficients form the M = 0
subset of the j × j = 1 × 1 set of
Clebsch-Gordan Coeffts.
Several authors have built TBM into models,
eg. G. Altarelli and F. Feruglio, hep-ph/0512103
I. de Medeiros Varzielas, S. King and G. Ross, hep-ph/0512313
E. Ma PRD 73:057304, 2006 etc. (A4, D(27), Σ(81) groups etc.)
Paul Harrison
University of Warwick
Birmingham, 30th April 2008
15
The Flavour problem
• The spectra of masses and mixings are today’s analogues of the Lyman, Balmer
and Paschen series. They show tantalising structure and yet are unexplained, and
inexplicable within the prevailing (Standard) Model.
• The Wolfenstein and Tri-bimaximal formulae, provide correct (within exptl. errors)
mathematical descriptions of the observed systematics of the mixing spectra,
without any explanation at all.
While based on less precise data, are somewhat akin to the Rydberg formula.
The data demand explanation, but the SM has nothing to say about them.
Paul Harrison
University of Warwick
Birmingham, 30th April 2008
16
Origin of Masses and Mixings in the SM
• In SM, they have common origin in fermion-Higgs “Yukawa couplings” (multiplied
by Higgs Vevs).
• Will focus on leptons – consider 3 families of leptonic SU (2) doublets:

ψa = 
νa
ℓ−
a

,


νb
,

ψb =
ℓ−
b

ψc = 
νc
ℓ−
c


Each transforms among itself under SU (2) weak transformations in usual way.
Paul Harrison
University of Warwick
Birmingham, 30th April 2008
16
Origin of Masses and Mixings in the SM
• In SM, they have common origin in fermion-Higgs “Yukawa couplings” (multiplied
by Higgs Vevs).
• Will focus on leptons – consider 3 families of leptonic SU (2) doublets:

ψa = 
νa
ℓ−
a

,


νb
,

ψb =
ℓ−
b

ψc = 
νc
ℓ−
c


Each transforms among itself under SU (2) weak transformations in usual way.
• Define also three-component vectors in family-space:
 
 
 
ψa
νa
ℓ−
 
 
 a
 
 
 
ψ =  ψb  with ν =  νb  ℓ = ℓ−

 
 
 b
ψc
νc
ℓ−
c
Paul Harrison
University of Warwick
Birmingham, 30th April 2008
17
Origin of Masses and Mixings (contd.)
• Generalise earlier Yukawa couplings to 3 families:
LY uk = (ψ L · φ) Yℓ ℓR + (ψ L · φ̃) Yν ν R + H.C.
Paul Harrison
University of Warwick
Birmingham, 30th April 2008
17
Origin of Masses and Mixings (contd.)
• Generalise earlier Yukawa couplings to 3 families:
LY uk = (ψ L · φ) Yℓ ℓR + (ψ L · φ̃) Yν ν R + H.C.
• After SSB, Dirac mass terms appear in Lagrangian:
Lmass = ℓ̄L Mℓ ℓR + ν̄ L MD ν R + H.C.
• Mℓ = Yℓ · (vev) and MD = Yν · (vev), are arbitrary, 3 × 3 Dirac mass matrices.
Paul Harrison
University of Warwick
Birmingham, 30th April 2008
17
Origin of Masses and Mixings (contd.)
• Generalise earlier Yukawa couplings to 3 families:
LY uk = (ψ L · φ) Yℓ ℓR + (ψ L · φ̃) Yν ν R + H.C.
• After SSB, Dirac mass terms appear in Lagrangian:
Lmass = ℓ̄L Mℓ ℓR + ν̄ L MD ν R + H.C.
• Mℓ = Yℓ · (vev) and MD = Yν · (vev), are arbitrary, 3 × 3 Dirac mass matrices.
• Charged-current weak interaction (same basis) is diagonal:
Lcc = g(ℓ̄L . ν L )W + + H.C.
-
W
-
la
1
Paul Harrison
University of Warwick
νa
Birmingham, 30th April 2008
18
Origin of Masses and Mixing (contd.)
−1
• For Majorana neutrinos, light ν mass matrix is Mν = MDT MM
aj MD (complex
symmetric).
Paul Harrison
University of Warwick
Birmingham, 30th April 2008
18
Origin of Masses and Mixing (contd.)
−1
• For Majorana neutrinos, light ν mass matrix is Mν = MDT MM
aj MD (complex
symmetric).
• Useful to work with Hermitian mass matrices. Define:
– for charged leptons: L = Mℓ Mℓ† OR L = Mℓ URℓ etc.
– for neutrinos: N = Mν Mν† OR N = MD MD† OR N = MD URν etc.
Paul Harrison
University of Warwick
Birmingham, 30th April 2008
18
Origin of Masses and Mixing (contd.)
−1
• For Majorana neutrinos, light ν mass matrix is Mν = MDT MM
aj MD (complex
symmetric).
• Useful to work with Hermitian mass matrices. Define:
– for charged leptons: L = Mℓ Mℓ† OR L = Mℓ URℓ etc.
– for neutrinos: N = Mν Mν† OR N = MD MD† OR N = MD URν etc.
• Diagonalise: Uν N Uν† = Dν → eigenvalues (eg. masses-squared), “physical” states.
Paul Harrison
University of Warwick
Birmingham, 30th April 2008
18
Origin of Masses and Mixing (contd.)
−1
• For Majorana neutrinos, light ν mass matrix is Mν = MDT MM
aj MD (complex
symmetric).
• Useful to work with Hermitian mass matrices. Define:
– for charged leptons: L = Mℓ Mℓ† OR L = Mℓ URℓ etc.
– for neutrinos: N = Mν Mν† OR N = MD MD† OR N = MD URν etc.
• Diagonalise: Uν N Uν† = Dν → eigenvalues (eg. masses-squared), “physical” states.
• Diagonalisation different for L and N ⇒ mixing, ie.
m
+
Lcc = g (ℓ̄ m
L UM N S ν L )W + H.C.
-
W
-
li
Ui α
να
where UM N S ≡ Uℓ Uν† is the lepton mixing matrix. NB. Analogous for quarks.
Paul Harrison
University of Warwick
Birmingham, 30th April 2008
19
Jarlskog (ie. Weak Basis) Covariance
• However L and N are not observable. Under a simultaneous change of weak basis:
L′ = UJ LUJ† and N ′ = UJ N UJ†
have same eigenvalues and same mixing matrix:
†
′
†
UM
=
U
(U
ℓ
NS
J UJ )Uν = UM N S
• ie. Masses and mixing angles are “Jarlskog-invariant”, even though mass matrices
transform.
Paul Harrison
University of Warwick
Birmingham, 30th April 2008
19
Jarlskog (ie. Weak Basis) Covariance
• However L and N are not observable. Under a simultaneous change of weak basis:
L′ = UJ LUJ† and N ′ = UJ N UJ†
have same eigenvalues and same mixing matrix:
†
′
†
UM
=
U
(U
ℓ
NS
J UJ )Uν = UM N S
• ie. Masses and mixing angles are “Jarlskog-invariant”, even though mass matrices
transform.
• New basis is just as good a starting point...
• Allows, eg. freedom to “push” mixing into neutrinos or charged leptons
Paul Harrison
University of Warwick
Birmingham, 30th April 2008
19
Jarlskog (ie. Weak Basis) Covariance
• However L and N are not observable. Under a simultaneous change of weak basis:
L′ = UJ LUJ† and N ′ = UJ N UJ†
have same eigenvalues and same mixing matrix:
†
′
†
UM
=
U
(U
ℓ
NS
J UJ )Uν = UM N S
• ie. Masses and mixing angles are “Jarlskog-invariant”, even though mass matrices
transform.
• New basis is just as good a starting point...
• Allows, eg. freedom to “push” mixing into neutrinos or charged leptons
• Jarlskog urges (1985): “important results can’t be frame-dependent” and reiterates
(hep-ph/0606050): “you should be able to formulate it in an invariant form”
• We adopt Jarlskog’s prescription as a principle!
Paul Harrison
University of Warwick
Birmingham, 30th April 2008
20
Historical Attempts to Constrain Fermion Masses and Mixings
• Inspired by phenomenological relationships in quark mixing, eg.
r
r
md
md
;
tan θcb ≃ 0.04 ≃
tan θC ≃ 0.23 ≃
ms
mb
and common origin of masses and mixings. Many attempts made to relate them.
• Best known is by H. Fritzsch

0
a

2

M 3 =  a∗
0

0
b∗
from 1977 and 1978: mass matrices:



0
0
α
0



 ∗


− 31
M = α
b ;
0
β 



c
0
β∗
γ
• Predicts (after approximations) above θC relation, but also
r
r
ms
mc
−
≃ 0.09,
(now excluded).
tan θcb >
mb
mt
Paul Harrison
University of Warwick
Birmingham, 30th April 2008
21
How does Fritzsch Ansatz work?
• Mass matrices encode 10 pieces of information (6 masses and 4 mixing
parameters) using only 6 magnitudes and one phase, =⇒ 3 constraints.
• Spawned many refinements, some phenomenologically successful.
• Modern models appeal to “flavour symmetries” to enforce zeroes or other
“textures”.
• Many build-in supersymmetry and/or complicated Higgs structures etc.
Paul Harrison
University of Warwick
Birmingham, 30th April 2008
21
How does Fritzsch Ansatz work?
• Mass matrices encode 10 pieces of information (6 masses and 4 mixing
parameters) using only 6 magnitudes and one phase, =⇒ 3 constraints.
• Spawned many refinements, some phenomenologically successful.
• Modern models appeal to “flavour symmetries” to enforce zeroes or other
“textures”.
• Many build-in supersymmetry and/or complicated Higgs structures etc.
• However, generally assume that ∃ a “special” basis, in which the mass matrices
take some “particular” form (which limits number of parameters to < 10).
• “Specialness” of this basis is never explained.
• Since 1905, we have been alert to excessive reliance on a given frame or basis.
• We seek, instead, a Jarskog-invariant description of mixing.
Paul Harrison
University of Warwick
Birmingham, 30th April 2008
22
Flavour-Symmetric Jarlskog Invariants
• “Usual” observables are not flavour-symmetric. eg. the electron mass, me ,
(electron label) and |Ue3 | (flavour and mass labels).
• Jarlskog showed how to write flavour-symmetric invariant observables.
Paul Harrison
University of Warwick
Birmingham, 30th April 2008
22
Flavour-Symmetric Jarlskog Invariants
• “Usual” observables are not flavour-symmetric. eg. the electron mass, me ,
(electron label) and |Ue3 | (flavour and mass labels).
• Jarlskog showed how to write flavour-symmetric invariant observables.
• Examples are (in an arbitrary weak-basis, primes dropped):
Tr(L2 ) = m2e + m2µ + m2τ ;
Tr(L) = me + mµ + mτ ;
Tr(L3 ) = m3e + m3µ + m3τ
which form a complete set.
Paul Harrison
University of Warwick
Birmingham, 30th April 2008
22
Flavour-Symmetric Jarlskog Invariants
• “Usual” observables are not flavour-symmetric. eg. the electron mass, me ,
(electron label) and |Ue3 | (flavour and mass labels).
• Jarlskog showed how to write flavour-symmetric invariant observables.
• Examples are (in an arbitrary weak-basis, primes dropped):
Tr(L2 ) = m2e + m2µ + m2τ ;
Tr(L) = me + mµ + mτ ;
Tr(L3 ) = m3e + m3µ + m3τ
which form a complete set.
• All traces of polynomials of L and N are invariant (and FS), eg.:
Tr(L′ N ′ ) = Tr(U LU † U N U † ) = Tr(U LN U † ) = Tr(LN U † U ) = Tr(LN )
etc.
• But typically depend in a complicated way on masses and mixing matrix elements.
Paul Harrison
University of Warwick
Birmingham, 30th April 2008
23
The Jarlskogian and Plaquette Invariance
Jarlskog’s CP -violating invariant:
∗
∗
J = Im(Uαi Uαj
Uβi
Uβj )
Paul Harrison
University of Warwick

∗
Ue3
Ue1 Ue2



 ∗

Uµ1 Uµ2 Uµ3 


Uτ 1 Uτ 2 Uτ 3
Birmingham, 30th April 2008
23
The Jarlskogian and Plaquette Invariance
Jarlskog’s CP -violating invariant:
∗
∗
J = Im(Uαi Uαj
Uβi
Uβj )

∗
Ue3
Ue1 Ue2



 ∗

Uµ1 Uµ2 Uµ3 


Uτ 1 Uτ 2 Uτ 3
Fascinating properties:
• Parameterises CP violation
• Does not depend on which plaquette is used
Paul Harrison
University of Warwick
Birmingham, 30th April 2008
23
The Jarlskogian and Plaquette Invariance
Jarlskog’s CP -violating invariant:
∗
∗
J = Im(Uαi Uαj
Uβi
Uβj )

∗
Ue1 Ue2
Ue3



 ∗

Uµ1 Uµ2 Uµ3 


Uτ 1 Uτ 2 Uτ 3
Fascinating properties:
• Parameterises CP violation
• Does not depend on which plaquette is used
• Simply related to the mass matrices:
J = −i
Det[L, N ]
2L∆ N∆
L∆ , N∆ are traces of polynomials in L and N .
Paul Harrison
University of Warwick
Birmingham, 30th April 2008
24
Are there Other Plaquette Invariants?
Paul Harrison
University of Warwick
Birmingham, 30th April 2008
24
Are there Other Plaquette Invariants?
Yes!
Paul Harrison
University of Warwick
Birmingham, 30th April 2008
24
Are there Other Plaquette Invariants?
Yes!
• J samples information uniformly across U
- it is f lavour − symmetric (FS).
Paul Harrison
University of Warwick
Birmingham, 30th April 2008
24
Are there Other Plaquette Invariants?
Yes!
• J samples information uniformly across U
- it is f lavour − symmetric (FS).
• Clearly, any function of the Uαi , symmetrised over all flavour labels,
and reduced to a function of only elements of a single plaquette
is plaquette-invariant.
Paul Harrison
University of Warwick
Birmingham, 30th April 2008
24
Are there Other Plaquette Invariants?
Yes!
• J samples information uniformly across U
- it is f lavour − symmetric (FS).
• Clearly, any function of the Uαi , symmetrised over all flavour labels,
and reduced to a function of only elements of a single plaquette
is plaquette-invariant.
• Working with observables, find that, like J, FS functions of the mixing matrix can
always be expressed as simple functions of the mass matrices.
• Will introduce an elemental set
– can be used for the flavour-symmetric description of any mixing scheme.
Paul Harrison
University of Warwick
Birmingham, 30th April 2008
25
The S3ℓ × S3ν Flavour Permutation Group
• 6 perms. of ℓ flavour indices and 6 of ν “flavour” labels (ie. ν mass eigenstate
indices) constitute the S3ℓ × S3ν Flavour Permutation Group (FPG).
Paul Harrison
University of Warwick
Birmingham, 30th April 2008
25
The S3ℓ × S3ν Flavour Permutation Group
• 6 perms. of ℓ flavour indices and 6 of ν “flavour” labels (ie. ν mass eigenstate
indices) constitute the S3ℓ × S3ν Flavour Permutation Group (FPG).
• Introduce the observable P matrix:


|Ue1 |2 |Ue2 |2 |Ue3 |2




P = |Uµ1 |2 |Uµ2 |2 |Uµ3 |2 


|Uτ 1 |2 |Uτ 2 |2 |Uτ 3 |2
– Parameterises mixing (up to the sign of J).
Paul Harrison
University of Warwick
Birmingham, 30th April 2008
25
The S3ℓ × S3ν Flavour Permutation Group
• 6 perms. of ℓ flavour indices and 6 of ν “flavour” labels (ie. ν mass eigenstate
indices) constitute the S3ℓ × S3ν Flavour Permutation Group (FPG).
• Introduce the observable P matrix:


|Ue1 |2 |Ue2 |2 |Ue3 |2




P = |Uµ1 |2 |Uµ2 |2 |Uµ3 |2 


|Uτ 1 |2 |Uτ 2 |2 |Uτ 3 |2
– Parameterises mixing (up to the sign of J).
– Transforms as a (reducible) 3 × 3 (natural representation) of FPG.
– Rows and columns each sum to unity (a magic square)
⇒ completely specified by elements of any P -plaquette.
– Each P -plaquette transforms as (irreducible) 2 × 2 of FPG.
Paul Harrison
University of Warwick
Birmingham, 30th April 2008
26
Singlets Under FPG?
• J is prototype FS observable - invariant under even members of the FPG;
flips sign under odd members; ie. 1 × 1.
Paul Harrison
University of Warwick
Birmingham, 30th April 2008
26
Singlets Under FPG?
• J is prototype FS observable - invariant under even members of the FPG;
flips sign under odd members; ie. 1 × 1.
• Search for other singlets: 1 × 1, 1 × 1 etc.
– Find simple polynomials of elements of P
– (anti-)symmetrise over flavour labels
Paul Harrison
University of Warwick
Birmingham, 30th April 2008
26
Singlets Under FPG?
• J is prototype FS observable - invariant under even members of the FPG;
flips sign under odd members; ie. 1 × 1.
• Search for other singlets: 1 × 1, 1 × 1 etc.
– Find simple polynomials of elements of P
– (anti-)symmetrise over flavour labels
• Simple representation theory→
– 1st order in P :
∃ no non-trivial singlets
– 2nd order:
one each of 1 × 1 and 1 × 1
– 3rd order:
one each of all four singlets
– ≥ 4th order:
multiple instances of each
• Will stay at ≤ 3rd order. Clearly four are sufficient.
Paul Harrison
University of Warwick
Birmingham, 30th April 2008
27
Elemental Set of FS Mixing Observables
Define themselves, up to normalisation (and “offset” in 1 × 1 case):
1 X
(Pαi )2 − 1 ]
G= [
2 αi
X
3 X
3
(Pαi )2 ] + 1
C= [
(Pαi ) −
2 αi
αi
F = DetP
1 X
A=
(Lγk )3
18 γk
where Lγk = (Pαi + Pβj − Pβi − Pαj ).
Paul Harrison
University of Warwick
Birmingham, 30th April 2008
27
Elemental Set of FS Mixing Observables
Define themselves, up to normalisation (and “offset” in 1 × 1 case):
1 X
(Pαi )2 − 1 ]
G= [
2 αi
X
3 X
3
(Pαi )2 ] + 1
(Pαi ) −
C= [
2 αi
αi
where Lγk = (Pαi + Pβj − Pβi − Pαj ).
• F and A need no offset (they are anti-symmetric).
– Reach extremum for no mixing
– 0 for trimaximal mixing
• All normalised to maximum value = 1 (no mixing).
• G and C offset to zero for maximal mixing
F = DetP
1 X
A=
(Lγk )3
18 γk


1 0 0




P = 0 1 0


0 0 1

1
3

P =  31

1
3
Paul Harrison
University of Warwick
1
3
1
3
1
3

1
3
1

3
1
3
Birmingham, 30th April 2008
28
Properties and Values
Observable
Order
Symmetry:
Theor.
Exptl. Range
Exptl. Range
Name
in P
S3ℓ × S3ν
Range
for Leptons
for Quarks
F
2
1×1
(−1, 1)
(−0.14, 0.12)
(0.893, 0.896)
G
2
1×1
(0, 1)
(0.15, 0.23)
(0.898, 0.901)
A
3
1×1
(−1, 1)
(−0.065, 0.052)
(0.848, 0.852)
C
3
1×1
1
, 1)
(− 27
(−0.005, 0.057)
(0.848, 0.852)
Properties and values of FS observables. Experimentally allowed ranges estimated
(90% CL) from compilations of current experimental results (neglect any correlations
between the input quantities).
Paul Harrison
University of Warwick
Birmingham, 30th April 2008
29
FSMOs in Terms of Mass Matrices
fm := Lm − 1 Tr(Lm )
Define reduced (ie. traceless) powers of mass matrices: L
3
m ).
g
(similarly for N
Now define Jarlskog-invariant:
fm N
fn ),
Temn := Tr(L
m, n = 1, 2.
Te Completely equivalent to P (for known lepton masses).
Paul Harrison
University of Warwick
Birmingham, 30th April 2008
29
FSMOs in Terms of Mass Matrices
fm := Lm − 1 Tr(Lm )
Define reduced (ie. traceless) powers of mass matrices: L
3
m ).
g
(similarly for N
Now define Jarlskog-invariant:
fm N
fn ),
Temn := Tr(L
m, n = 1, 2.
Te Completely equivalent to P (for known lepton masses).
Find:
Det Te
F ≡ Det P = 3
;
L ∆ N∆
Paul Harrison
University of Warwick
cf. J = −i
Det[L, N ]
2L∆ N∆
Birmingham, 30th April 2008
29
FSMOs in Terms of Mass Matrices
fm := Lm − 1 Tr(Lm )
Define reduced (ie. traceless) powers of mass matrices: L
3
m ).
g
(similarly for N
Now define Jarlskog-invariant:
fm N
fn ),
Temn := Tr(L
m, n = 1, 2.
Te Completely equivalent to P (for known lepton masses).
Find:
Det Te
;
F ≡ Det P = 3
L ∆ N∆
Temn Tepq Lmp N nq
G=
;
2
(L∆ N∆ )
Det[L, N ]
cf. J = −i
2L∆ N∆
(nqs)
(mpr)
Temn Tepq Ters LC,A NC,A
C, A =
(L∆ N∆ )KC,A
m ).
fm (N
g
The L (N ) are simple functions of traces of L
Paul Harrison
University of Warwick
KC (KA ) = 2(3).
Birmingham, 30th April 2008
30
Application: Flavour-symmetric Descriptions of Mixing
MNS Matrix:
ν1
e



|U | ∼ µ 

τ
⇒ F = 0,
Paul Harrison
C = 0,
A = 0,
√
2/ 6
√
1/ 6
√
1/ 6
ν2
√
1/ 3
√
1/ 3
√
1/ 3
ν3
ǫ
√
1/ 2
√
1/ 2





G = 61 (1 − 3ǫ2 )2 .
University of Warwick
Birmingham, 30th April 2008
30
Application: Flavour-symmetric Descriptions of Mixing
MNS Matrix:
ν1
e



|U | ∼ µ 

τ
⇒ F = 0,
C = 0,
A = 0,
√
2/ 6
√
1/ 6
√
1/ 6
ν2
√
1/ 3
√
1/ 3
√
1/ 3
ν3
ǫ
√
1/ 2
√
1/ 2





G = 61 (1 − 3ǫ2 )2 .
Where is flavour-symmetry?
Paul Harrison
University of Warwick
Birmingham, 30th April 2008
31
Where is Flavour Symmetry?
• It is spontaneously broken
Paul Harrison
University of Warwick
Birmingham, 30th April 2008
31
Where is Flavour Symmetry?
• It is spontaneously broken
• There exist 36 equally valid “solutions” of the TBM form, related to each other by
allowed permutations of the rows and columns of the mixing matrix.
• eg.
ν1
e



|U | ∼ µ 

τ
Paul Harrison
√
1/ 3
√
1/ 3
√
1/ 3
ν2
√
2/ 6
√
1/ 6
√
1/ 6
University of Warwick
ν3
ǫ
√
1/ 2
√
1/ 2





Birmingham, 30th April 2008
32
FS Descriptions of Several Mixing Schemes
Particular mixing schemes and their corresponding descriptions in terms of constraints
on FS mixing observables, and symmetries.
Mixing Ansatz
F
G
C
A
No Mixing
Tribimaximal Mixing∗
Trimaximal Mixing
S3 Group Mixing∗
Two Equal P -Rows∗
Two Equal P -Columns
Altarelli-Feruglio∗
Tri-χmaximal Mixing∗
Tri-φmaximal Mixing∗
Bi-maximal Mixing
1
0
0
0
0
0
0
0
0
0
1
1
0
0
0
–
–
1
0
0
–
0
0
0
0
–
0
Paul Harrison
1
6
0
–
–
–
–
–
1
6
1
8
6G−1
8
0
0
1
− 32
University of Warwick
Corresponding
Symmetries
CP
Dem., µ-τ , CP
Dem., µ-τ
Democracy
e.g. µ-τ
e.g. 1-2
µ-τ , CP
Dem., µ-τ
Dem., CP
CP , µ-τ , 1-2
18J 2
B
D
0
0
0
0
0
0
0
–
0
0
0
0
0
1
6
–
–
–
0
–
0
0
1√
12 3
0
–
–
0
–
–
–
0
Birmingham, 30th April 2008
32
FS Descriptions of Several Mixing Schemes
Particular mixing schemes and their corresponding descriptions in terms of constraints
on FS mixing observables, and symmetries.
Mixing Ansatz
F
G
C
A
No Mixing
Tribimaximal Mixing∗
Trimaximal Mixing
S3 Group Mixing∗
Two Equal P -Rows∗
Two Equal P -Columns
Altarelli-Feruglio∗
Tri-χmaximal Mixing∗
Tri-φmaximal Mixing∗
Bi-maximal Mixing
1
0
0
0
0
0
0
0
0
0
1
1
0
0
0
–
–
1
0
0
–
0
0
0
0
–
0
1
6
0
–
–
–
–
–
1
6
1
8
6G−1
8
0
0
1
− 32
Corresponding
Symmetries
CP
Dem., µ-τ , CP
Dem., µ-τ
Democracy
e.g. µ-τ
e.g. 1-2
µ-τ , CP
Dem., µ-τ
Dem., CP
CP , µ-τ , 1-2
18J 2
B
D
0
0
0
0
0
0
0
–
0
0
0
0
0
1
6
–
–
–
0
–
0
0
1√
12 3
0
–
–
0
–
–
–
0
• Setting for example, F 2 + C 2 + A2 + J 2 = 0 provides a FS, JI constraint on the
mass matrices (and, by extrapolation, on a combination of vevs times Yukawas).
Paul Harrison
University of Warwick
Birmingham, 30th April 2008
32
FS Descriptions of Several Mixing Schemes
Particular mixing schemes and their corresponding descriptions in terms of constraints
on FS mixing observables, and symmetries.
Mixing Ansatz
F
G
C
A
No Mixing
Tribimaximal Mixing∗
Trimaximal Mixing
S3 Group Mixing∗
Two Equal P -Rows∗
Two Equal P -Columns
Altarelli-Feruglio∗
Tri-χmaximal Mixing∗
Tri-φmaximal Mixing∗
Bi-maximal Mixing
1
0
0
0
0
0
0
0
0
0
1
1
0
0
0
–
–
1
0
0
–
0
0
0
0
–
0
1
6
0
–
–
–
–
–
1
6
1
8
6G−1
8
0
0
1
− 32
Corresponding
Symmetries
CP
Dem., µ-τ , CP
Dem., µ-τ
Democracy
e.g. µ-τ
e.g. 1-2
µ-τ , CP
Dem., µ-τ
Dem., CP
CP , µ-τ , 1-2
18J 2
B
D
0
0
0
0
0
0
0
–
0
0
0
0
0
1
6
–
–
–
0
–
0
0
1√
12 3
0
–
–
0
–
–
–
0
• Setting for example, F 2 + C 2 + A2 + J 2 = 0 provides a FS, JI constraint on the
mass matrices (and, by extrapolation, on a combination of vevs times Yukawas).
• Such constraints may be ”easily” implemented in BSM models, and allow the
spontaneous breaking of flavour symmetry.
Paul Harrison
University of Warwick
Birmingham, 30th April 2008
33
A Further Application
• What feature do quark and lepton

∼1


|V
|
∼
 ∼λ
CKM
mixing matrices have in common?

3
∼
λ
• Each has at least one “small” ele
ment.

• What is the flavour-symmetric ex
|UM N S | ∼ 

pression of this?
Paul Harrison
University of Warwick
√
2/ 6
√
1/ 6
√
1/ 6
∼λ
∼1
∼ λ2
√
1/ 3
√
1/ 3
√
1/ 3
∼ λ3


2 
∼λ 

∼1
ǫ
√
1/ 2
√
1/ 2





Birmingham, 30th April 2008
33
A Further Application
• What feature do quark and lepton

∼1


|V
|
∼
 ∼λ
CKM
mixing matrices have in common?

3
∼
λ
• Each has at least one “small” ele
ment.

• What is the flavour-symmetric ex
|UM N S | ∼ 

pression of this?
√
2/ 6
√
1/ 6
√
1/ 6
∼λ
∼1
∼ λ2
∼ λ3


2 
∼λ 

∼1
√
1/ 3
√
1/ 3
√
1/ 3
ǫ
√
1/ 2
√
1/ 2





≥ 1 zero ⇒ CP-conservation =⇒ J = 0.
But need two constraints - find:
2A + F (F 2 − 2C − 1) = 0 and J = 0 =⇒ at least one zero
Paul Harrison
University of Warwick
Birmingham, 30th April 2008
34
A Further Application (Cont.)
∗
∗
• NB. Consider K-matrix: Kγk = Re(Uαi Uαj
Uβi
Uβj ) (cf. J).
Paul Harrison
University of Warwick
Birmingham, 30th April 2008
34
A Further Application (Cont.)
∗
∗
• NB. Consider K-matrix: Kγk = Re(Uαi Uαj
Uβi
Uβj ) (cf. J).
• Noticed that 2A + F (F 2 − 2C − 1) ≡ DetK
• Both MNS and CKM mixing matrices satisfy DetK = 0 (within errors)
Paul Harrison
University of Warwick
Birmingham, 30th April 2008
34
A Further Application (Cont.)
∗
∗
• NB. Consider K-matrix: Kγk = Re(Uαi Uαj
Uβi
Uβj ) (cf. J).
• Noticed that 2A + F (F 2 − 2C − 1) ≡ DetK
• Both MNS and CKM mixing matrices satisfy DetK = 0 (within errors)
• Conjecture: MNS and CKM constrained according to the same FS JI condition:
DetK = 0;
J small.
• =⇒ FS prediction: if DetK = 0, then as J → 0, at least one UT angle → 90◦ .
Paul Harrison
University of Warwick
Birmingham, 30th April 2008
34
A Further Application (Cont.)
∗
∗
• NB. Consider K-matrix: Kγk = Re(Uαi Uαj
Uβi
Uβj ) (cf. J).
• Noticed that 2A + F (F 2 − 2C − 1) ≡ DetK
• Both MNS and CKM mixing matrices satisfy DetK = 0 (within errors)
• Conjecture: MNS and CKM constrained according to the same FS JI condition:
DetK = 0;
J small.
• =⇒ FS prediction: if DetK = 0, then as J → 0, at least one UT angle → 90◦ .
For quarks: DetK = 0 =⇒ (90◦ − α) = ηλ2 = 1◦ ± 0.2◦
.
◦
cf. (90 − α) =
For leptons: DetK = 0 =⇒ (90◦ − δ) =
Paul Harrison
University of Warwick
◦+3◦
0 −7◦
|Ue3 |
√
2
experimentally.
< 8◦ (good for CP V ).
∼
Birmingham, 30th April 2008
35
Summary
• Experimentally, flavour parameters appear anything but arbitrary
• SM does not, and cannot predict them - profoundly unsatisfactory
Paul Harrison
University of Warwick
Birmingham, 30th April 2008
35
Summary
• Experimentally, flavour parameters appear anything but arbitrary
• SM does not, and cannot predict them - profoundly unsatisfactory
• Minimal disruption to SM suggests models of masses and mixings should be
Jarlskog (weak-basis) invariant
Paul Harrison
University of Warwick
Birmingham, 30th April 2008
35
Summary
• Experimentally, flavour parameters appear anything but arbitrary
• SM does not, and cannot predict them - profoundly unsatisfactory
• Minimal disruption to SM suggests models of masses and mixings should be
Jarlskog (weak-basis) invariant
• Have defined S3ℓ × S3ν -symmetric, JI, mass and/or mixing observables.
• “Simplest” set defines itself up to normalisation
Paul Harrison
University of Warwick
Birmingham, 30th April 2008
35
Summary
• Experimentally, flavour parameters appear anything but arbitrary
• SM does not, and cannot predict them - profoundly unsatisfactory
• Minimal disruption to SM suggests models of masses and mixings should be
Jarlskog (weak-basis) invariant
• Have defined S3ℓ × S3ν -symmetric, JI, mass and/or mixing observables.
• “Simplest” set defines itself up to normalisation
• Remarkably, leptonic mixing is consistent with 3 of these = 0!
Paul Harrison
University of Warwick
Birmingham, 30th April 2008
35
Summary
• Experimentally, flavour parameters appear anything but arbitrary
• SM does not, and cannot predict them - profoundly unsatisfactory
• Minimal disruption to SM suggests models of masses and mixings should be
Jarlskog (weak-basis) invariant
• Have defined S3ℓ × S3ν -symmetric, JI, mass and/or mixing observables.
• “Simplest” set defines itself up to normalisation
• Remarkably, leptonic mixing is consistent with 3 of these = 0!
• Can use these to implement FS and JI constraints on flavour observables, and
make testable conjectures
Paul Harrison
University of Warwick
Birmingham, 30th April 2008
35
Summary
• Experimentally, flavour parameters appear anything but arbitrary
• SM does not, and cannot predict them - profoundly unsatisfactory
• Minimal disruption to SM suggests models of masses and mixings should be
Jarlskog (weak-basis) invariant
• Have defined S3ℓ × S3ν -symmetric, JI, mass and/or mixing observables.
• “Simplest” set defines itself up to normalisation
• Remarkably, leptonic mixing is consistent with 3 of these = 0!
• Can use these to implement FS and JI constraints on flavour observables, and
make testable conjectures
• More speculatively, such quantities could be used to implement models of
spontaneous flavour violation.
Paul Harrison
University of Warwick
Birmingham, 30th April 2008