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A(4) Family Symmetry and Quark-Lepton Unification
Michal Malinský
UK Neutrino network meeting, Oxford, November 29 2006
School of Physics and Astronomy, University of Southampton
S.F.King, M.M., JHEP11(2006)071, ArXiv: hep-ph/0608021
S.F.King, M.M., ArXiv: hep-ph/0610250
S. Antusch, S.F.King, M.M., work in progress
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Michal Malinský 2UK Neutrino network meeting, Oxford, November
29 2006
Outline
• General comments on SUSY flavour models
• Hints from the neutrino sector
• Sample SO(3) x Pati-Salam model
• A(4) flavour symmetry and quark-lepton unification
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Michal Malinský 3UK Neutrino network meeting, Oxford, November
29 2006
General comments on (SUSY) flavour models
In (MS)SM, the flavour physics is governed by the Yukawa
couplings (and SSB sector)
SU(3)c ⊗ SU(2)L ⊗ U(1)YMatter multiplets:
QiL =
(
u1 u2 u3d1 d2 d3
)i
L
= (3, 2, +1/3)
U ciL =(
uc1 uc2 u
c3
)i
L= (3, 1,−4/3)
DciL =(
dc1 dc2 d
c3
)i
L= (3, 1, +2/3)
LiL =
(
νll
)i
L
= (1, 2,−1)
N ciL =(
νc)i
L= (1, 1, 0)
EciL =(
lc)i
L= (1, 1, +2)
Yukawa interactions:LY ! Y
ijU Q
iL
TC−1U c
jLHu + Y
ijD Q
iL
TC−1Dc
jLHd+
+YijN L
iL
TC
−1N
cjLHu + Y
ijE L
iL
TC
−1E
cjLHd + h.c.
Majorana sector:LM ! Y
ij∆
LiL
TC
−1L
jL∆ + M
ijν
ciL
TC
−1ν
cjL + h.c.
Quark and charged lepton masses:
M iju ∝ YijU vu, M
ijd ∝ Y
ijD vd, M
ijl ∝ Y
ijE vd mν
.= Y∆〈∆〉 − Y
TN M
−1YNv
2u
type-II type-I
Seesaw mechanism:
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Michal Malinský 4UK Neutrino network meeting, Oxford, November
29 2006
In both cases, the extra symmetries must be badly broken at the
electroweak scale
Extra symmetries proposed to constrain the Yukawa (and SSB)
structures
General comments on (SUSY) flavour models
Horizontal symmetries
Q1L Q2
L Q3
L
Ec1L E
c2L E
c3L
Uc1L U
c2L U
c3L
Nc1L N
c2L N
c3L
L1
L L2
L L3
L
Dc1L D
c2L D
c3L
Yukawa entries correlated
!QL
!UcL
!DcL
!NcL
!EcL
!LL
...SU(3), SO(3)
Extended gauge symmetries
{
Q1L Q2
L Q3
L
Ec1L E
c2L E
c3L
Uc1L U
c2L U
c3L
Nc1L N
c2L N
c3L
L1
L L2
L L3
L
Dc1L D
c2L D
c3L
{
QcL
LcL
Yukawa matrices correlated
...SU(3)c ⊗ SU(2)L ⊗ SU(2)R ⊗ U(1)B−L,
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Michal Malinský 5UK Neutrino network meeting, Oxford, November
29 2006
The flavour symmetry breaking (usually driven by flavour
symmetry Higgs fields - flavons) must be well under control to lead
to correct spectra and mixing patterns.
General comments on (SUSY) flavour models
It is typically transmitted to the matter sector via higher
order vertices with flavour symmetry breaking flavon VEVs:
h
FL FcL
Φ
.
.
.
.
.
.
spectra mixing
Typically driven by flavour symmetry breaking scale(s)
Typically driven by vacuum alignment
Realistic model = flavour symmetry + FS breaking + vacuum
alignment mechanism
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Michal Malinský 6UK Neutrino network meeting, Oxford, November
29 2006
General comments on (SUSY) flavour models
In SUSY, there is an extra piece of information coming from FCNC
and CPV which get naturaly enhanced due to the off-diagonalities in
the soft sector
Usually, the larger the flavour symmetry multiplets the better
the control over the Kähler potential and the soft sector.
SUSY flavour problem
µ → eγ, b → sγ, . . .
M.Bando,N.Maekawa, Prog.Theor.Phys. 106 (2001)
1255S.F.King,M.Oliveira, Phys.Rev. D63 (2001)
095004F.-S.Ling,P.Ramond, Phys.Lett. B543 (2002)
29F.-S.Ling,P.Ramond, Phys.Rev. D67 (2003)
115010W.Grimus,L.Lavoura, Eur.Phys.J. C28 (2003) 123G.L.Kane et
al., JHEP 08 (2005) 083 T.Blazek,S.Raby,K.Tobe, Phys.Rev.D62 (2000)
055001S.Raby, Phys.Lett.B561 (2003) 119S.F.King,G.G.Ross,
Phys.Lett.B520 (2001) 243S.F.King,G.G.Ross, Phys.Lett.B574 (2003)
239R.Barbieri et al., [hep-ph/9901228]. S.Antusch,S.F. King,
Nucl.Phys.B705 (2005) 239
C.Hagedorn et al.,Phys. Rev. D74 (2006) 025007W.Grimus et al.,
JHEP07 (2004) 078R.Dermisek,S.Raby, Phys. Lett. B622 (2005)
327R.Dermisek et al., Phys. Rev.D74 (2006) 035011
C.Hagedorn,M.Lindner,R.N.Mohapatra, JHEP06 (2006)
042K.S.Babu,E.Ma,J.W.F.Valle, Phys.Lett.B552 (2003) 207
U(1), SU(2), SU(3), SO(3)
SUSY CP problem
EDMs, mixing,...K − K
D(3), D(4), D(5), S(4), A(4) . . .
Discrete symmetries fine for the vacuum alignment
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Michal Malinský 7UK Neutrino network meeting, Oxford, November
29 2006
Hints from the neutrino sector
indicates correlations among all three lepton families.
U ∼
√
2
3
1√
30
−1√
6
1√
3
1√
2
−1√
6
1√
3−
1√
2
Tri-bimaximal mixing in the lepton sector:
P.F.Harrison,D.Perkins,W. G. Scott, Phys.Lett.B530, 167 (2002)
Both SUSY constraints and lepton mixing tend to call for maximal
flavor symmetries !
Similar correlations arise for the neutrino Yukawa entries. If
RH neutrinos happen to be sufficiently hierarchical and the charged
lepton sector mixing negligible (in the basis in which RH neutrinos
are diagonal), the sequential dominance mechanism gives the
tri-bimaximal pattern automatically for
YνLR ∼
0 b .
a b .
−a b c
S. F. King, Nucl. Phys. B576 (2000) 85 S. F. King, JHEP 09
(2002) 011
a2
M1!
b2
M2!
c2
M3
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Michal Malinský 8UK Neutrino network meeting, Oxford, November
29 2006
A sample SO(3) x Pati-Salam model
〈!φ23〉 ∼
0
v
−v
, 〈!φ123〉 ∼
ṽ
ṽ
ṽ
, 〈!φ3〉 ∼
.
.
V
leads to the desired neutrino Yukawa matrix provided
Issues to be addressed:
RH neutrino sector should be essentially diagonal
Charged lepton sector should not spoil the T-B lepton mixing
The quark and lepton mass hierarchies and CKM mixing should be
accommodated
The vacuum alignment mechanism
W leadingY =1
My23 !F .!φ23F
c1h +
1
My123 !F .!φ123F
c2h +
1
My3 !F .!φ3F
c3h + . . .
For example
{
Q1L Q2
L Q3
L
L1
L L2
L L3
L
Ec1L E
c2L E
c3L
Uc1L U
c2L U
c3L
Nc1L N
c2L N
c3L
Dc1L D
c2L D
c3L
{
F c1 , Fc
2 , Fc
3 ≡!F ≡
(3; 4, 1, 2)3 × (1; 4, 2, 1)
← SO(3) →
←P
S→
PS = SU(4)C ⊗ SU(2)L ⊗ SU(2)RS. F. King, JHEP 08 (2005) 105S. F.
King, M.M., JHEP 11 (2006) 071
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Michal Malinský 9UK Neutrino network meeting, Oxford, November
29 2006
A sample SO(3) x Pati-Salam model
the standard vacuum alignment mechanism very complicated and
quite expensive !!!
The model works quite well, but:
Y fLR =
0 y123εf123
y12εf12
εf3
y23εf23
y123εf123
+ CfyGJ ε̃f23
σ ỹ23(ε̃23)2ε3−y23ε
f23
y123εf123
+ CfyGJ ε̃f23
σ y3εf3
σ ≡ 〈Σ〉/Mf
εfx ≡ |〈 "φx〉|/Mf
yx ∼ O(1)
Cu,d,l,ν = −2, 1, 3, 0
are the Clebsches to accommodate the proper mass hierarchies
ε23, ε123, ε12 ! ε̃23 < ε3 ∼ O(1)
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Michal Malinský 10UK Neutrino network meeting, Oxford, November
29 2006
A(4) x Pati-Salam model
Discrete symmetries can help with the huge vacuum degeneracy of
the continuous case !!!
Example: SO(3) and A(4) invariants that can be built out of
triplets:
A(4)SO(3)
quadratic:
cubic:
quartic:
φ.φ φ.φ
(φ × χ).ψ,(φ × χ).ψ (φ ∗ χ).ψ
(φ.φ)2, (φ × ψ)2, . . . (φ.φ)2,3∑
i=1
φiφiφiφi, (φ × ψ)2, . . .
A(4) is the symmetry group of tetraheadron, i.e. a discrete
subgroup of SO(3)Equivalently, it is a group of even permutations
of 4 objects.
B.Adhikary et al.,Phys. Lett. B638 (2006) 345E.Ma,
H.Sawanaka,M.Tanimoto,hep-ph/0606103G.Altarelli,F.Feruglio, Nucl.
Phys. B720 (2005) 64I.de Medeiros
Varzielas,S.F.King,G.G.Ross,hep-ph/0512313 L.Lavoura,H.Kuhbock,
hep-ph/0610050. G. Altarelli,F. Feruglio,Y.Lin, hep-ph/0610165.
E.Ma,G.Rajasekaran,Phys.Rev.D64(2001)113012K.S.Babu,E.Ma,J.W.F.Valle,Phys.Lett.B552(2003)207M.
Hirsch et al., Phys. Rev. D69 (2004) 093006S.-L.
Chen,M.Frigerio,E.Ma, Nucl.Phys.B724(2005)423E. Ma,Phys. Rev. D72
(2005)037301
Benefits of a discrete subgroup in the game:
Invariants of the continuous case remain intact and new terms
are allowed
The extra terms break explicitly the original continuous
symmetry
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Michal Malinský 11UK Neutrino network meeting, Oxford, November
29 2006
Example:
The (single) flavon scalar potential
V ! −M2φ(φ†φ) + Λ(φ†φ)2
Isocurves in 2D projection
λ > 0 : 〈!φ〉 ∼
1
1
1
λ < 0 : 〈!φ〉 ∼
0
0
1
or perms.
!φ3, !φ123 VEVs almost for free !!!
On the other hand, it might be difficult to get such simple
structures from the F-terms.
I. de Medeiros Varzielas, S. F. King, and G. G. Ross,
hep-ph/0607045.
However, higher order D-terms can naturally lead to a set of
such extra quartic terms in the effective potential.
+λ φ†iφiφ
†iφi + . . .
can in A(4) case contain terms like:
A(4) x Pati-Salam model
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Michal Malinský 12UK Neutrino network meeting, Oxford, November
29 2006
Slight modification of the previous SO(3) model:
〈!φ1〉 ∼
v10
0
Vacuum:
〈!φ2〉 ∼
0
v20
〈!φ2〉 ∼
0
0
V3
〈!φ123〉 ∼
vvv
As we have seen these structures are easy to get !
〈!φ23〉 ∼
0
v23−v23
〈!̃φ23〉 ∼
0
V23−V23
How to get these ?
The Yukawa sector remains quite similar to that of the SO(3)
case, but the vacuum alignment is very simple !
A(4) x Pati-Salam model
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Michal Malinský 13UK Neutrino network meeting, Oxford, November
29 2006
Obtaining : 〈!φ23〉 ∼
0
v−v
V ! −M223|φ23|2
+ λ123|φ†123
.φ23|2
+ λ1|φ†1.φ23|
2+ . . .
Minimized for orthogonal to〈!φ23〉 〈!φ123〉 Minimized for
orthogonal to〈!φ23〉 〈!φ1〉
S. F. King, M.M., hep-ph/0610250
Virtues of the vacuum alignment in the discrete case :
Simplicity
Extra constraints on the Kähler potential
A(4) x Pati-Salam model
Handle on the soft SUSY breaking sector (?)
Work in progress...
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Michal Malinský 14UK Neutrino network meeting, Oxford, November
29 2006
Conclusions
• SUSY flavour models strongly constrained in both Yukawa and
soft sectors
• SUSY flavour and CP problems call for maximal symmetries
• Models with discrete subgroups of continuous symmetries
provide for simple vacuum alignment mechanisms
• Work in progress
Thanks for your kind attention !