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Leptonic CP violation theory
C. Hagedorn
CP3 -Origins, SDU, Odense, Denmark
XXVII International Conference on Neutrino Physics and Astrophysics,
04.07.-09.07.2016, London, UK
– p. 1/30
Low energy CP phases in the lepton sector
Parametrization
UP M N S = Ũ diag(1, eiα/2 , ei(β /2+δ ) )
with

c12 c13
s13 e−iδ
s12 c13

iδ
Ũ = 
−s
c
−
c
s
s
e
12
23
12
23
13

s12 s23 − c12 c23 s13 eiδ
iδ
c12 c23 − s12 s23 s13 e
−c12 s23 − s12 c23 s13 eiδ
s23 c13
c23 c13




and sij = sin θij , cij = cos θij
Experimental status
• first indications for δ ∼
3
2
π
(Bergström et al. (’16), Capozzi et al. (’16))
• no constraints on Majorana phases α and β
– p. 2/30
Prediction of CP phases with a flavor symmetry
• framework: 3 copies of charged leptons & Majorana neutrinos
• impose flavor symmetry Gf on space of 3 lepton generations
• Gf is non-abelian, finite and discrete
• Gf must be broken at low energies and we assume residual
symmetries Ge (Gν ) in the charged lepton (neutrino) sector
(Blum/H/Lindner (’07), Lam (’07,’08))
֒→ form of charged lepton mass matrix me & Majorana mass
matrix mν restricted by Ge and Gν
֒→ contributions Ue & Uν to lepton mixing UP M N S are fixed
– p. 3/30
Prediction of CP phases with a flavor symmetry
Gf
ւ
ց
charged leptons
neutrinos
Ge
Gν = Z2 × Z2
Ue
Uν
ց
ւ
UP M N S = Ue† Uν
[Masses do not play a role in this approach.]
– p. 4/30
Prediction of CP phases with a flavor symmetry
UP M N S = Ue† Uν
• 3 unphysical phases are removed by Ue → Ue Ke
• neutrino masses are made real and positive through Uν → Uν Kν
• permutations of columns of Ue , Uν are possible Ue,ν → Ue,ν Pe,ν
⇓
Predictions
Mixing angles up to exchange of rows/columns
Dirac phase δ up to π
Majorana phases undetermined
– p. 5/30
Prediction of CP phases with a flavor symmetry
• surveys of groups Gf , Ge and Gν
(H/Meroni/Vitale (’13), King/Neder/Stuart (’13), Joshipura/Patel (’13), Holthausen et al. (’12),
Lam (’12), Fonseca/Grimus (’14), Talbert (’14))
have shown that
all patterns have trimaximal column (sin2 θ12 & 1/3) and
Dirac phase δ = 0, π,
if mixing angles are accommodated well
• studies with one massless neutrino, (partially) degenerate
neutrino masses or Dirac neutrinos also exist
(Joshipura/Patel (’13,’14), King/Ludl (’16), Hernandez/Smirnov (’13), Esmaili/Smirnov (’15))
– p. 6/30
Prediction of CP phases with a flavor symmetry
For less symmetry, e.g. Gν = Z2 , Dirac phase is less constrained
(Hernandez/Smirnov (’12))
• example with Gf = PSL(2, 7), Ge = Z7 and Gν = Z2
thick line: δ = 0
dashed line: δ = π/4
dotted line: δ = π/2
– p. 7/30
Prediction of CP phases with a flavor symmetry
With corrections Dirac phase is less constrained
(Marzocca et al. (’11), Petcov (’14), Shimizu et al. (’14), Ballett et al. (’14), Girardi et al. (’14,’15,’16))
• assume
e
e
UP M N S = Ue† Ψ Uν with Ue = R23 (θ23
) R12 (θ12
)
and Ψ phases
and Uν is tribimaximal (TBM), bimaximal (BM), golden ratio
(GR) mixings as well as hexagonal mixing (HEX/HG)
ν
ν
= 0 and θ23
= π/4)
(all have in common θ13
ν
• relation of δ, lepton mixing angles θij and θ12
follows
2
cos δ =
tan θ23 sin θ12 +
sin2 θ13 cos2 θ12
tan θ23
2
− sin
ν
θ12
tan θ23 +
sin2 θ13
tan θ23
sin 2θ12 sin θ13
– p. 8/30
Prediction of CP phases with a flavor symmetry
With corrections Dirac phase is less constrained
(Ballett et al. (’14))
– p. 9/30
Prediction of CP phases with a flavor symmetry
With corrections Dirac phase is less constrained
(Girardi et al. (’14))
Other possible corrections originate from RG running (Zhang/Zhou (’16)).
– p. 10/30
Prediction of CP phases with a flavor and a CP
symmetry
• impose also CP symmetry in fundamental theory
• CP symmetry acts non-trivially on the flavor space
(Grimus/Rebelo (’95))
its action is φi → Xij φ⋆j and XX † = XX ⋆ = 1
• in a theory with Gf and CP certain conditions have to be fulfilled
(Feruglio/H/Ziegler (’12,’13), Holthausen et al. (’12), Chen et al. (’14))
• prediction of Majorana phases becomes possible
(Feruglio/H/Ziegler (’12,’13))
for mν invariant under CP
X mν X = m⋆ν
֒→ Uν is further constrained
– p. 11/30
Prediction of CP phases with a flavor and a CP
symmetry
• well-known example: µτ reflection symmetry
muon (tau) neutrino ↔ tau (muon) antineutrino
(Harrison/Scott (’02,’04), Grimus/Lavoura (’03))
which leads – in the charged lepton mass basis – to
sin θ23 = cos θ23
and
sin 2θ12 sin θ13 cos δ = 0
and trivial Majorana phases
– p. 12/30
Prediction of CP phases with a flavor and a CP
symmetry
(Feruglio/H/Ziegler (’12,’13))
Gf and CP
ւ
ց
charged leptons
neutrinos
Ge
Gν = Z2 × CP
Uν = Ων R(θ)Kν
Ue
ց
ւ
UP M N S = Ue† Ων R(θ)Kν
[Masses do not play a role in this approach.]
– p. 13/30
Prediction of CP phases with a flavor and a CP
symmetry
UP M N S = Ue† Ων R(θ)Kν
• 3 unphysical phases are removed by Ue → Ue Ke
• UP M N S contains one free parameter θ
• permutations of rows and columns of UP M N S possible
⇓
Predictions
Mixing angles and all CP phases are predicted
in terms of one free parameter θ only,
up to permutations of rows/columns
– p. 14/30
Example of predictions of low energy CP
phases: ∆(3 n2 ), ∆(6 n2 ) and CP
(H/Meroni/Molinaro (’14); see also Ding et al. (’14))
∆(3 n2 ), ∆(6 n2 ) and CP
ւ
ց
charged leptons
neutrinos
Ge = Z3
Gν = Z2 × CP
Ue
Uν = Ων R(θ)Kν
ց
ւ
UP M N S = Ue† Ων R(θ)Kν
four different types of mixing patterns with different characteristics
– p. 15/30
Example of predictions of low energy CP
phases: ∆(3 n2 ), ∆(6 n2 ) and CP
Case 3 b.1)
(H/Meroni/Molinaro (’14))
• first column is fixed via choice of residual Z2 symmetry
Z2 (m) in neutrino sector
• in particular, solar mixing angle constrains m to be m ≈
n
2
• free parameter θ is fixed by reactor mixing angle
• for m =
n
2
we find lower limit on CP violation via Dirac phase
| sin δ| & 0.71
and both Majorana phases α, β depend on X = X(s) only
| sin α| = | sin β| = | sin 6 φs | with φs =
πs
and s = 0, ..., n−1
n
– p. 16/30
Example of predictions of low energy CP
phases: ∆(3 n2 ), ∆(6 n2 ) and CP
Case 3 b.1) and n = 8, m = 4:
some viable choices of s
(H/Meroni/Molinaro (’14))
s
sin2 θ13
sin2 θ12
sin2 θ23
sin δ
s=1
0.0220
0.318
0.579
0.936
0.0220
0.318
0.421
−0.936
s=2
0.0216
0.319
0.645
−0.739
1
s=4
0.0220
0.318
0.5
∓1
0
sin α = sin β
√
−1/ 2
√
−1/ 2
– p. 17/30
Low energy CP phases in 0νββ decay
Case 3 b.1) and n = 8, m = 4
(H/Molinaro (’16))
Other type of constraints on α, β and 0νββ decay comes from neutrino mass sum rules (Altarelli/Feruglio (’05), King et al. (’13), Gehrlein et al. (’15,’16)).
– p. 18/30
Further approaches with CP symmetries
• many more flavor symmetries have been combined with CP
(Di Iura/H/Meloni (’15), Li/Ding (’15), Ballett et al. (’15), Rong (’16), Yao/Ding (’16), ...)
• study of more general CP symmetries (Everett et al. (’15))
• two CP symmetries in neutrino sector (Chen et al. (’14))
• textures of CP transformations (Chen et al. (’16))
• considerable efforts in building models in recent past
(Altarelli, Antusch, Branco, Centelles Chulia, Chen, Chu, Dasgupta, de Medeiros
Varzielas, Ding, Everett, Feruglio, Gehrlein, Girardi, Gonzalez Felipe, Grimus, H, He,
Joaquim, King, Lavoura, Luhn, Mahanthappa, Machado, Meloni, Meroni, Mohapatra, Neder, Nishi, Päs, Pascoli, Petcov, Rodejohann, Schumacher, Serodio, Shimizu,
Smirnov, Spinrath, Srivastava, Stuart, Tanimoto, Valle, Vien, Xu, Yamamoto, Ziegler, ...)
– p. 19/30
High energy CP phases in the lepton sector
• context: type-I seesaw mechanism
• 3 right-handed (RH) Majorana neutrinos Ni
• their masses Mi are taken to be
1012 GeV . Mi . 1014 GeV
• integrating Ni out, light neutrino masses are
mν ≈ −mD MR−1 mTD with mD = YD hHi
What can we say about CP phases?
– p. 20/30
High energy CP phases in the lepton sector
What can we say about CP phases?
• mass matrix MR contains six complex parameters
• CP phases in MR not related to low energy ones
• but they are relevant for leptogenesis (Fukugita/Yanagida (’86))
– p. 21/30
High energy CP phases in the lepton sector
What can we say about CP phases?
• mass matrix MR contains six complex parameters
• CP phases in MR not related to low energy ones
• but they are relevant for leptogenesis (Fukugita/Yanagida (’86))
simplest scenario (unflavored)
YB ∼ 10−3 ǫ η with ǫ CP asymmetry , η efficiency factor
compare to observations:
YB = (8.65 ± 0.09) × 10−11
(Planck (’15))
֒→ if 10−3 . η . 1, we need 10−4 & ǫ & 10−7
– p. 22/30
Example of predictions of high energy CP
phases: ∆(3 n2 ), ∆(6 n2 ) and CP
How to apply Gf and CP here?
(H/Molinaro (’16))
• MR is invariant under Gν ,
while YD is invariant under Gf and CP
• me is invariant under Ge
– p. 23/30
Example of predictions of high energy CP
phases: ∆(3 n2 ), ∆(6 n2 ) and CP
How to apply Gf and CP here?
(H/Molinaro (’16))
• MR is invariant under Gν ,
while YD is invariant under Gf and CP
• me is invariant under Ge
What can we say about neutrino masses and lepton mixing?
for YD ∝ 1 in flavor space we find
mi ∝
1
Mi
and Uν = UR = Ων R(θ)Kν
and together with me diagonal we get
UP M N S = Uν = UR = Ων R(θ)Kν
– p. 24/30
Example of predictions of high energy CP
phases: ∆(3 n2 ), ∆(6 n2 ) and CP
How to apply Gf and CP here?
(H/Molinaro (’16))
• MR is invariant under Gν ,
while YD is invariant under Gf and CP
• me is invariant under Ge
Result: no CP asymmetry ǫ is produced
(Jenkins/Manohar (’08), Bertuzzo et al. (’09), H/Molinaro/Petcov (’09),
Aristizabal Sierra et al. (’09))
– p. 25/30
Example of predictions of high energy CP
phases: ∆(3 n2 ), ∆(6 n2 ) and CP
How to apply Gf and CP here?
(H/Molinaro (’16))
• MR is invariant under Gν ,
while YD is invariant under Gf and CP
• me is invariant under Ge
Needed: corrections δYD ∝ κ to YD
that are invariant under Ge
– p. 26/30
Example of predictions of high energy CP
phases: ∆(3 n2 ), ∆(6 n2 ) and CP
How to apply Gf and CP here?
(H/Molinaro (’16))
• MR is invariant under Gν ,
while YD is invariant under Gf and CP
• me is invariant under Ge
Needed: corrections δYD ∝ κ to YD
that are invariant under Ge
Results: ǫ ∝ κ2 – explains small value of ǫ
sign of ǫ can be fixed because of determined phases
– p. 27/30
Example of predictions of high energy CP
phases: ∆(3 n2 ), ∆(6 n2 ) and CP
Case 3 b.1) and n = 8, m = 4
(H/Molinaro (’16))
For flavored leptogenesis in theories with flavor & CP symmetry
see (Mohapatra/Nishi (’15), Chen et al. (’16), Yao/Ding (’16)).
– p. 28/30
Comments on CP violation in the quark sector
• flavor symmetries can successfully explain Cabibbo angle
(Blum/H/Lindner (’07), Lam (’07))
π
≈ 0.2225 from dihedral groups D7 and D14
|Vus | = sin
14
• however, describing all quark mixing angles well is difficult
(Holthausen/Lim (’13), Araki et al. (’13), Ishimori et al. (’14), Yao/Ding (’15), de
Medeiros Varzielas et al. (’16))
• smaller quark mixing angles likely from corrections
• CP violation in the quark sector is still a mystery
– p. 29/30
Conclusions – leptonic CP violation
• flavor symmetries alone can predict Dirac phase
• if in addition a CP symmetry is present, also both
Majorana phases can be predicted
• in a scenario with RH neutrinos we constrain high energy
CP phases as well and thus the sign of the baryon asymmetry of the Universe
Thank you for your attention.
– p. 30/30

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