1
.
Flavour Permutation Symmetry and Quark Mixing:
Is the Unitarity Triangle Right?
Paul Harrison∗
University of Warwick
SuperB Workshop, Isola d’Elba
1st June 2008
∗
With Dan Roythorne and Bill Scott, PLB 657 (2007) 210, arXiv:0709.1439 and
arXiv:0805.3440
Paul Harrison
University of Warwick
1st June 2008
2
.
Outline of Talk
• The Flavour Problem
• Plaquette Invariance and Flavour Permutation Symmetry
• Flavour Symmetric Mixing Observables
• Applications
• Conclusions
Paul Harrison
University of Warwick
1st June 2008
3
The Flavour Problem
• SM has 20 (22) low energy flavour parameters (out of 25(27))
• They show tantalising structure, eg.
d
u

O(1)


|VCKM | ∼ c  O(λ)

t
O(λ3 )
ν1
e



|UM N S | ∼ µ 

τ
√
2/ 6
√
1/ 6
√
1/ 6
s
O(λ)
O(1)
O(λ2 )
ν2
√
1/ 3
√
1/ 3
√
1/ 3
b
O(λ3 )



O(λ2 ) 

O(1)
+
W
qi
ν3
<
∼ 0.2
√
1/ 2
√
1/ 2





-
W
-
li
• Are not predictable in the SM - profoundly unsatisfactory.
Paul Harrison
qα
Vi α
University of Warwick
Ui α
να
1st June 2008
4
Masses and Mixings have Common Origin
• in Quark and Lepton mass matrices: M 2 , M− 1 , Mℓ , Mν
3
3
• Each is product of Higgs vev and Yukawa coupling matrix in Lagrangian
Paul Harrison
University of Warwick
1st June 2008
4
Masses and Mixings have Common Origin
• in Quark and Lepton mass matrices: M 2 , M− 1 , Mℓ , Mν
3
3
• Each is product of Higgs vev and Yukawa coupling matrix in Lagrangian
• Will work with their Hermitian Squares.
– eg. for neutrinos: N = Mν Mν†
– for charged leptons: L = Mℓ Mℓ†
– for charged 2/3 quarks: H 2 = M 2 M 2† etc.
3
Paul Harrison
3
3
University of Warwick
1st June 2008
4
Masses and Mixings have Common Origin
• in Quark and Lepton mass matrices: M 2 , M− 1 , Mℓ , Mν
3
3
• Each is product of Higgs vev and Yukawa coupling matrix in Lagrangian
• Will work with their Hermitian Squares.
– eg. for neutrinos: N = Mν Mν†
– for charged leptons: L = Mℓ Mℓ†
– for charged 2/3 quarks: H 2 = M 2 M 2† etc.
3
3
3
• Diagonalise Uν N Uν† = Dν → eigenvalues (ie. masses-squared)
• Diagonalisation different for L and N ⇒ mixing, ie. UM N S ≡ Uℓ Uν† .
Analogous for quarks: UCKM ≡ U− 1 U 2† .
3
Paul Harrison
3
University of Warwick
1st June 2008
5
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
1st June 2008
5
The Jarlskogian and Plaquette Invariance
Jarlskog’s CP -violating invariant:
∗
∗
J = Im(Uαi Uαj
Uβi
Uβj )
Fascinating properties:

∗
Ue3
Ue1 Ue2



 ∗

Uµ1 Uµ2 Uµ3 


Uτ 1 Uτ 2 Uτ 3
• Parameterises CP violation
• Does not depend on which plaquette is used
Paul Harrison
University of Warwick
1st June 2008
5
The Jarlskogian and Plaquette Invariance
Jarlskog’s CP -violating invariant:
∗
∗
J = Im(Uαi Uαj
Uβi
Uβj )
Fascinating properties:

∗
Ue3
Ue1 Ue2



 ∗

Uµ1 Uµ2 Uµ3 


Uτ 1 Uτ 2 Uτ 3
• Parameterises CP violation
• Does not depend on which plaquette is used
• Simply related to the mass matrices:
Det[L, N ]
J = −i
2L∆ N∆
L∆ , N∆ are traces of polynomials in L and N .
Paul Harrison
University of Warwick
1st June 2008
6
Are there Other Plaquette Invariants?
Paul Harrison
University of Warwick
1st June 2008
6
Are there Other Plaquette Invariants?
Yes!
Paul Harrison
University of Warwick
1st June 2008
6
Are there Other Plaquette Invariants?
Yes!
• J samples information uniformly across U
- it is f lavour − symmetric (FS).
Paul Harrison
University of Warwick
1st June 2008
6
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
1st June 2008
6
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
1st June 2008
7
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
1st June 2008
7
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
1st June 2008
7
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
1st June 2008
8
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
1st June 2008
8
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
1st June 2008
8
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
1st June 2008
9
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
1st June 2008
9
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
1st June 2008
10
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
1st June 2008
11
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
1st June 2008
11
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
cf. J = −i
University of Warwick
Det[L, N ]
2L∆ N∆
1st June 2008
11
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∆ )nC,A
m ).
fm (N
g
The L (N ) are simple functions of traces of L
Paul Harrison
University of Warwick
nC (nA ) = 2(3).
1st June 2008
12
Application: Flavour-symmetric Descriptions of Mixing
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
10
ν1
e



|U | ∼ µ 

τ
⇒ F = 0,
Paul Harrison
ν2
1
ν3

2
3
4
10
10
10
Distance to Reactor (m)
10
5
Atmospheric
√
2/ 6
√
1/ 6
√
1/ 6
√
1/ 3
√
1/ 3
√
1/ 3
ǫ
√
1/ 2
√
1/ 2
C = 0,
A = 0,
G = 61 (1 − 3ǫ2 )2 . Where is flavour-symmetry?




University of Warwick
1st June 2008
13
Where is Flavour Symmetry?
• It is spontaneously broken
Paul Harrison
University of Warwick
1st June 2008
13
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





1st June 2008
14
Another Application:
Partially Unified FS Description of Quark and Lepton Mixings
Unified Description of quark and lepton mixings is desirable.
Paul Harrison
University of Warwick
1st June 2008
14
Another Application:
Partially Unified FS Description of Quark and Lepton Mixings
Unified Description of quark and lepton mixings is desirable.
What feature do quark and lepton mixing matrices have in common?
Paul Harrison
University of Warwick
1st June 2008
14
Another Application:
Partially Unified FS Description of Quark and Lepton Mixings
Unified Description of quark and lepton mixings is desirable.
What feature do quark and lepton mixing matrices have in common?

∼1


|V
|
∼
 ∼λ
CKM
• Each has at least one “small” ele
∼ λ3
ment.
• What is the flavour-symmetric expression of this?
Paul Harrison



|UM N S | ∼ 

University of Warwick
√
2/ 6
√
1/ 6
√
1/ 6
∼λ
∼1
∼ λ2
√
1/ 3
√
1/ 3
√
1/ 3
∼ λ3



∼ λ2 

∼1
ǫ
√
1/ 2
√
1/ 2





1st June 2008
14
Another Application:
Partially Unified FS Description of Quark and Lepton Mixings
Unified Description of quark and lepton mixings is desirable.
What feature do quark and lepton mixing matrices have in common?

∼1


|V
|
∼
 ∼λ
CKM
• Each has at least one “small” ele
∼ λ3
ment.
• What is the flavour-symmetric expression of this?



|UM N S | ∼ 

√
2/ 6
√
1/ 6
√
1/ 6
∼λ
∼1
∼ λ2
√
1/ 3
√
1/ 3
√
1/ 3
∼ λ3



∼ λ2 

∼1
ǫ
√
1/ 2
√
1/ 2





≥ 1 zero ⇒ CP-conservation =⇒ J = 0. But need two constraints.
Paul Harrison
University of Warwick
1st June 2008
15
FS Condition for One Mixing Zero
After some work, find
=⇒ 2A + F (F 2 − 2C − 1) = 0 and J = 0
• Manifestly FS, but not “neat”
Paul Harrison
University of Warwick
1st June 2008
15
FS Condition for One Mixing Zero
After some work, find
=⇒ 2A + F (F 2 − 2C − 1) = 0 and J = 0
• Manifestly FS, but not “neat”
∗
∗
• Now, consider K-matrix: Kγk = Re(Uαi Uαj
Uβi
Uβj ) (cf. J).
• Familiar from CP conserving parts of interference terms in penguin dominated
rates
Paul Harrison
University of Warwick
1st June 2008
15
FS Condition for One Mixing Zero
After some work, find
=⇒ 2A + F (F 2 − 2C − 1) = 0 and J = 0
• Manifestly FS, but not “neat”
∗
∗
• Now, consider K-matrix: Kγk = Re(Uαi Uαj
Uβi
Uβj ) (cf. J).
• Familiar from CP conserving parts of interference terms in penguin dominated
rates
• In fact 2A + F (F 2 − 2C − 1) ≡ 54 Det K
Paul Harrison
University of Warwick
1st June 2008
15
FS Condition for One Mixing Zero
After some work, find
=⇒ 2A + F (F 2 − 2C − 1) = 0 and J = 0
• Manifestly FS, but not “neat”
∗
∗
• Now, consider K-matrix: Kγk = Re(Uαi Uαj
Uβi
Uβj ) (cf. J).
• Familiar from CP conserving parts of interference terms in penguin dominated
rates
• In fact 2A + F (F 2 − 2C − 1) ≡ 54 Det K
So
Det K = 0 and J = 0
is condition we want
Paul Harrison
University of Warwick
1st June 2008
16
FS Condition for One Small Element
• For one small element, relax (slightly) one condition or both.
• For quark mixing, know that J 6= 0 (but small)
Paul Harrison
University of Warwick
1st June 2008
16
FS Condition for One Small Element
• For one small element, relax (slightly) one condition or both.
• For quark mixing, know that J 6= 0 (but small)
• How about DetK?
< 3 × 10−7
|Det Kq /(Det K)max | ∼
• Leptons also consistent with Det K small or zero.
Paul Harrison
University of Warwick
1st June 2008
16
FS Condition for One Small Element
• For one small element, relax (slightly) one condition or both.
• For quark mixing, know that J 6= 0 (but small)
• How about DetK?
< 3 × 10−7
|Det Kq /(Det K)max | ∼
• Leptons also consistent with Det K small or zero.
• Conjecture: MNS and CKM constrained according to the same FS condition:
DetK = 0;
J small.
Unified, partial description of quark and lepton mixings.
Paul Harrison
University of Warwick
1st June 2008
17
Predictions
• DetK = 0; J small =⇒ as J → 0, at least one UT angle → 90◦ .
• Get precise prediction for one unitarity angle in terms of other mixing elements.
• Doesn’t tell us which angle, but the data do.
Paul Harrison
University of Warwick
1st June 2008
17
Predictions
• DetK = 0; J small =⇒ as J → 0, at least one UT angle → 90◦ .
• Get precise prediction for one unitarity angle in terms of other mixing elements.
• Doesn’t tell us which angle, but the data do.
• For quarks, predict:
− cos α ≃ (90◦ − α) = ηλ2 = 1◦ ± 0.2◦
.
Paul Harrison
.
◦
cf. (90◦ − α) = 0◦+3
−7◦ experimentally.
University of Warwick
1st June 2008
17
Predictions
• DetK = 0; J small =⇒ as J → 0, at least one UT angle → 90◦ .
• Get precise prediction for one unitarity angle in terms of other mixing elements.
• Doesn’t tell us which angle, but the data do.
• For quarks, predict:
− cos α ≃ (90◦ − α) = ηλ2 = 1◦ ± 0.2◦
.
.
◦
cf. (90◦ − α) = 0◦+3
−7◦ experimentally.
• “A sub-1◦ measurement is possible in B → ρρ and B → ππ individually”
(A. Bevan 5th SuperB workshop, Paris, 2007).
Paul Harrison
University of Warwick
1st June 2008
17
Predictions
• DetK = 0; J small =⇒ as J → 0, at least one UT angle → 90◦ .
• Get precise prediction for one unitarity angle in terms of other mixing elements.
• Doesn’t tell us which angle, but the data do.
• For quarks, predict:
− cos α ≃ (90◦ − α) = ηλ2 = 1◦ ± 0.2◦
.
◦
cf. (90◦ − α) = 0◦+3
−7◦ experimentally.
.
• “A sub-1◦ measurement is possible in B → ρρ and B → ππ individually”
(A. Bevan 5th SuperB workshop, Paris, 2007).
• For leptons, predict:
√
π < ◦
| cos δ| ≃ |90 − δ| = 2 2 sin θ13 sin (θ23 − ) ∼ 4
4
◦
Paul Harrison
University of Warwick
1st June 2008
18
Summary
• Have defined flavour-symmetric mass and/or mixing observables.
Paul Harrison
University of Warwick
1st June 2008
18
Summary
• Have defined flavour-symmetric mass and/or mixing observables.
• “Simplest” set defines itself up to normalisation
Paul Harrison
University of Warwick
1st June 2008
18
Summary
• Have defined flavour-symmetric 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
1st June 2008
18
Summary
• Have defined flavour-symmetric mass and/or mixing observables.
• “Simplest” set defines itself up to normalisation
• Remarkably, leptonic mixing is consistent with 3 of these = 0!
• Used them to implement FS constraints on flavour observables, and make testable
conjectures
Paul Harrison
University of Warwick
1st June 2008
18
Summary
• Have defined flavour-symmetric mass and/or mixing observables.
• “Simplest” set defines itself up to normalisation
• Remarkably, leptonic mixing is consistent with 3 of these = 0!
• Used them to implement FS constraints on flavour observables, and make testable
conjectures
• αCKM should be (89 ± 0.2)◦ - UT is almost right!
Paul Harrison
University of Warwick
1st June 2008
18
Summary
• Have defined flavour-symmetric mass and/or mixing observables.
• “Simplest” set defines itself up to normalisation
• Remarkably, leptonic mixing is consistent with 3 of these = 0!
• Used them to implement FS constraints on flavour observables, and make testable
conjectures
• αCKM should be (89 ± 0.2)◦ - UT is almost right!
• δleptons should be within 4◦ of 90◦ .
Paul Harrison
University of Warwick
1st June 2008