This document outlines a study on discrete flavor symmetries and CP violation in the lepton sector. It introduces discrete groups that can describe lepton mixing patterns and defines a consistent way to incorporate CP. Examples are discussed where geometrical CP violation can arise from mismatched symmetries between the charged lepton and neutrino sectors. The determination of leptonic mixing angles from global fits is also summarized.

1. CP and Discrete Flavour
Symmetries
based arXiv:1211.6953 in collaboration with Manfred Lindner and
Michael A. Schmidt – to appear in JHEP
Martin Holthausen
Max-Planck-Institut für Kernphysik
Heidelberg

2. Outline
! Introduction
! Lepton Mixing from Discrete Groups
! Consistent definition of CP
! CP and automorphisms
! Examples
! Geometrical CP Violation
! Conclusions

3. 13 and Leptonic CP Violation
• Recent Progess in Determination of Leptonic Mixing Angles
global fit m2
21 m2
31 sin2 ✓12 sin2 ✓23 sin2 ✓13
[Forero, Tortola, [10 5 eV2 ] [10 3 eV2 ] [10 1] [10 1] [10 2] [⇡]
Valle 2012]
best fit 7.62+.19
.19 2.55+.06
.09 3.20+.16
.17 6.13+.22
.40 2.46+.29
.28 0.8+1.2
.8
3 range 7.12 8.20 2.31 2.74 2.7 3.7 3.6 6.8 1.7 3.3 0 2
• Largish 13 neutrino oscillation parameters (for normal ordering of neutrino masses) adapted
Table 1: Global fit of established
from [17]. The errors of the best fit values indicate the one sigma ranges. In the global fit there are two nearly
• Hint for non-maximal 23
degenerate minima at sin2 ✓23 = 0.430+.031 , see Figure 1. Double
T2K
.030
• unknowns: Chooz
• Normal or Inverted Hierarchy? RENO
only the structure of flavor symmetry group and its remnant symmetries are assumed and we
DayaBay
• Majorana or Dirac (L-number conservation) ?
do not consider the breaking mechanism i.e. how the required vacuum alignment needed to
• Is 23 maximal?
achieve the remnant symmetries is dynamically realized.
• Is there CP violation in the lepton sector?
The PMNS matrix is defined as
• !CP
• Majorana Phases UPMNS = Ve† V⌫ ✓13 = 8.6+.44 (1).46
• are there sterile neutrinos that sizably mix with
and can be determined from the unitary matrices Ve and V⌫ satisfying
the active ones [talk by M. Lindner 2 weeks
T ago] † ⇤ 2 2 2 T

4. ˜
U = @ s12 c23 c12 s23 s13 e c12 c23 s12 s23 s13 e s23 c13 A . (13
s12 s23 c12 c23 s13 ei c12 s23 s12 c23 s13 ei c23 c13
13 and Leptonic CP Violation
shorthand notation sij = sin ✓ij and cij = cos ✓ij . The mixing angles ✓ij rang
⇡/2, while the Majorana phases ↵, as well as the Dirac phase take value
and 2⇡.
s.(10,11) we see that
⇤ ?
Xe UP M N S X⌫ = UP M N S (14
s.(8,9) are fulfilled, eq.(14) is simplified to [20]
= (
lskog invariant JCP [21]
⇥
)(
UP M N S,ij e ixi
x
⇤
sj = UP M N S,ij
)(
⇤
x ) (15
⇤ ⇤
JCP = Im UP M N S,11 UP M N S,13 UP M N S,31 UP M N S,33
1
= sin 2✓12 sin 2✓23 sin 2✓13 cos ✓13 sin (16
8
nish,3 • Largish 13 means CP violation can be observed in oscillations in not-so-distant
future (see W. Winter’s talk last week)
JCP = 0 , sin = 0 . (17

5. mixing amongst all three mass eigenstates. Therefore this symmetry cannot b
of the entire Lagrangianbrokenite )has, to be⇢(g⌫ )⌦⌫ hase⇢(g⌫di↵erent to di↵erent subgroups Ge G
Lagrangian but mixing to beof but entireunderstood as a consequenceG⌫ mismatched horizontal sym- a
certain it has †patterns can be di↵erent subgroups G to be broken subgroups G and
the to Lagrangian but it and of (with
to
⌦e ⇢(ge )⌦e Lagrangian but † broken be broken to di↵erent subgroups G e
of lepton and intersection) in respectively.lepton and 1 26–28]4 . Let (6) assume If
= ⇢(g diag ⌦⌫ has to= )
ection) in metries acting on the charged lepton andit chargedsectorsdiag neutrino sectors, respectively. e
the chargedthe entire neutrino sectors, the
Lepton mixing from discrete groups
1 trivial 0 neutrino If the[11–13;
fermions us
trivial intersection) inthat there is as (discrete) symmetry group G under which therespectively. If
the charged the charged lepton and sectors, respectively. If the
transform a leptoncos ✓ neutrino sin ✓
and
0 where ⇢(g) thisare diagonal unitary matrices. These conditions idetermine ⌦ sectors, to a
1 0 purposeintersection) in
trivial
for
diag
0 e neutrino e , ⌦⌫ up left-handed
f
C transform as B under a
transform A matrix K and 13 (✓, complete matrices P
1
0 C
i as doublets L = (⌫, e) transformflavour groupfaithful unitary 3-dimensional representation
p
diagonal lepton
p
phase
e ! ⇢(ge )e,
and U permutation
⌫ e,⌫ ⇢(g⌫ )⌫,
!
)=@ 0 1
ge 2 Ge , g⌫! ⇢(ge,)e,
e 2 G⌫ e,⌫ 0 A. (13)
⌫ ! ⇢(g⌫ )⌫, (4) ge 2 Ge , g⌫ 2 G⌫ ,
Gf
2 2
1
0 p2: G ! GL(3, ):
⇢
i
e i sin
therefore restrict the ✓ 0 matrices have to fulfil
cos ✓
arged lepton fmasses and we the symmetry to hold, ourselves to the abelian case.
p
2
metry to symmetry of (Me Me+) e ! ⇢(ge )e, fulfil⌦⌫ !)e, e,⌫ .)⌫, ! ⇢(g⌫ )⌫, Ge , ge residual ,⌫g⌫ 2 G⌫ ,
residual hold, the mass matrices to ! ⇢(ge ⇢(g ⌫
for have ⌦e,⌫ e ! e,⌫ Ke,⌫ P ⌫ mass ge 2 2 G , (7)
⌫ 2 Ge symmetry of M
ict ourselves to the case of Majorana ! ⇢(g)L, L neutrinos, which .implies that there
g2G f (3)
neutrino and charged mixing
lepton e )T Me Mand e )⇤
masses † we = M M † restrict ourselves M ⇢(g ) = M⌫ .
therefore
does not completely fixsymmetry to hold, thematrix yet, as the first fulfil third
the leptonic ⇢(g )T to the abelian c
to and
T
⇢(g )T M Me ⇢(gefor the e M †
† LH⇢(g
leptons 3-dim e ⇢(g
)⇤ to matrices ⇢(g and ⌫mass rep. and
are 2
ambiguities of the last equation, V (5) given⌫by ⌦⌫ . ⌫
⇢(g ) M ⇢(g ) = M . e e
plex eeigenvalueWe = Mhold, the ourselves⌫ (i) they all matrices satisfy ⇢(g⌫ ) which (ii) there is th
therefore have are unequal and e,⌫
It follows efrom Eqn. (5) thate up to the ⌫ ) matrices ⌫have to fulfil e,⌫
of furtherdata clearly shows to the case lepton masses neutrinos, = 1, implies that
and
for the symmetry correspondingmass
The experimental restrict eigenstatesthat beofrotated into each other
the samebe seen as
and the can
⌫
Majorana
This can det ⇢(g⌫ ) = 1. By further erequiringbe a non-abelian group would lead to a degenerate ma
er choose be a amongst allgroup would lead toto degenerate mass symmetryMajorana a symmetry)2
or G⌫ symmetry. To Tbe three⇢(ge )T theM⌫ ⇢(gethreeMe we⇢(g ) and they⇢(g⌫ ) be ⌫satisfy ⇢(g⌫
distinguishable therefore
Choosing eigenvalue of ⇤ matrices †
complex G M G † a )Therefore
cannot mass or mixing matrix
ng the tomixingnon-abelian a as their fix⇤eigenstates.†cannot beMthis ⌫spectrum, cannotT⌫ xZ2 ⌫ ) = M⌫ .=
completely representations the = haveand enlarge inequivalent one-dim
oup G⌫ is restrictedwe canefurtherbutT three M⇤ ⌦Z)2=⇢TZBy T Mdi↵erent ⇤subgroups=Z)and M⌫ .(with
⇢(gedecomposed⇢(g⇢ )M= † ⇢ e Me broken todecomposedS,U ⌫1 distinguishable Major
eG = to into three G 2 ⇢(gM
esentations cannot ⌦T =Z3e e
Gethe entireat e↵ecteof adding⇢(g ⇤ This scenario is not diag
of = T be
†
and the e
e e
det ⇢(g T three Ge
)Lagrangian⌦Tchoosehasinequivalent 2one-dimensionalto M ⇢(g⌫ = G ⌫
toM † ⌦⇤ = Klein group be=⇥1. and be†requiring determine 0
M 1 e into it M to ⌫
be M . To M0⌦⇤ ⇢ ⌫ )rep-
further able
nerator. Let us look Me the resentations of Ge or G⌫ .symmetry egenerator U 0with
0 e e e the diag ⌦e
0 1 is not the mixingbe a the case ofthe Klein group Z2 ⇥ Z2 .withisthecaseto determ
0 GgroupG ⌫ to restricted to be threegroup compatible to Ifbedegenerate m
e e 1e 0 of 0 1
the 1 0 three d
ionse of rows and columns) e4compatible with non-abelian group would lead To a @ fermions
neutrinos the or G is matrix from the distinguished
of G or Gtrivial intersection) inthe charged lepton and neutrino sectors, structure it
Choosing
⌫ . This scenario ⌫ respectively. able
A a
G = @ G⌫ a representations and
has topresentationtransform be inequivalentinvariant decomposed into is true for
betransform 0 their 1 asinWe non-abelian columns) G .mixing matrixa
⇢(T ) e or asto0 todiagonalhere
(up matrix the
0 in 1
Choosingdiagonal (onlypermutations27]. rowscannot bewhen convention samethree inequivalent0one-dim
A
1group would@lead by 27]. degenerate massAsp
0 offollowisthe presentation and conjugated [26; a 0from⇢(U ) = 0 structure i
⇢(S) =
to arbitrary phase
the group
1
llow the as
e all neutrinos and convention [26; 1 0 0singlets of ⌫ The0 0 1 0 1 0
necessary 0 cannotneutrinos transform as inequivalentinequivalent The same is true
1 0 to have all
matrix) and the resentations of G or Gdecomposed be is not to bring it into G⌫ . one-dimensio
phasing and permutation . freedom can into three singlets with the case of three
C scenario used compatible of the
Bbe smaller than Z . We can now determineform
onstheir representations 3 ⌘! ⇢(gebe 0 This ⇢(g⌫ )⌫, 3 ge (Z xZ emost 2 G⌫ , choice if mixing angles do not
as which shows⇢(U ) = U cannote 0 ⌫ 1 ! .
that Ge
2 , m2 , m but and analogously@ )e, showsA ⌫ that G cannot be G2 , g⌫ than (14)
22 smaller
2
diag(me µ ⌧ the charged e
(Z3 smallest choice, 2 ), can also be leptons for ⌦⌫ . From thesee group theoreticalgeneral Z3 . We can now determ
which considerations we
(4)
he continuous) matricesWe ehere.⌫ This misaligned non-commuting ⌫ that satisfy on27]. the Majorana three distin
resentations of thee4mixing⌦ follow the0 1matricesand ,conventiondependwith & case of s)
unitary G or G⌫via thatunitary 0 is not compatible masses
⌦ , 2 scenario
the satisfy
can thus determine the PMNS matrix presentation ⌦e ⌦
for the symmetry to hold, the mass matrices have to fulfil
in [26;
4 symmetries lead to
We †here follow the† presentation and convention in [26; 27]. †
⌦†e )⇤ )⌦ = e ⌦⌫ diag ,
⌦† ⇢(g M 1,⇢(g ⌫ )
value⌦e ⇢(ge )⌦e = ⇢(g⇢(gdiagMe Me ⇢(gU⇢(ge⌫)⌦=⌫⌦= ⇢(gand thus ⌫ ⇢(g⌫mixing) matrix (8)
e ) e )T , ⌦ the )⌦ = = M (6)
of ✓ to zero, ⌦U ⇢(U )⌦U =† diag( =1, ee M†1),e )anddiag ⇢(g⌫ )T M⌫⌫⇢(g⌫⇢(g⌫ )diag
⌫e PMNS †
2 ⌫. (5)
mixing matrix determined from symmetry
tri-bimaximal mixing ⇢(g)diag are diagonal unitary matrices. These conditions determine ⌦e , ⌦⌫ up t
Volkas where (TBM) form up to interchanging of rows/columns and
re [He, Keum,Choosing Geofmatrices.xZ2a non-abelian group2would lead to that it is not possible
diagonal 06;
up to07, 08; unitaryG rows and columns. It should not determine ⌦a ,degenerate mass spectrum,
Lam
a permutation or= G⌫ to=Z These conditions be surprising e ⌦⌫ phase to a
S,U
be diagonal up matrix
as and permutation matrix,q and permutation to predict lepton one-dimensional
0 as 1
2
to uniquely pin diagonal phase matrix Kbe decomposed into matrices Pe,⌫
Altarelli,Feruglio 05, representations cannot e,⌫ it is not
matrix Ke,⌫ theirdown the mixing matrices 2Pe,⌫p possible three inequivalentmasses in this rep-
1
Feruglio, 0
resentations of Ge or G⌫ . This scenario is3not compatible with the case of three distinguished
3
approach. B C G =S
Hagedorn,Toroop 11] †
⌘@ B p 1 1 ⌦e,⌫ ! ⌦e,⌫ Ke,⌫ Pe,⌫ . =Z2
1 C,
Let us now= ⌦efollow= Upresentation ande,⌫ . some interesting cases. We have0 (15) the ei sin ✓ 1
UPMNS try to U
4 ⌦ apply this machinery to p
the HPS
p
2 A
We here ⌦e,⌫ ! ⌦e,⌫ Ke,⌫ P convention in [26; 27].
6 3 seen that 0
(7)
cos↵✓
⌦
smallest residual It follows fromthe charged p up psector p given byofM Me = UTT |T@ = E V⇠ Z3 . given by ⌦
symmetry in Eqn. (5) that lepton1 the ambiguities UT the = BM 3
1 to 1
is a G last equation, = are 0 A
0 e,⌫ 1
tri-bimaximal 6 3 2 i
qn. (5) that upwhere can be seen 1 is given by 1 equation, Ve,⌫ are given by ⌦e,⌫ .✓ 0 cos ✓
We use a basis This the generator of the last
as
to the ambiguities 2 2
e sin
mixing (TBM) sin2 2 tri-maximal mixing (TM2)
onds to the mixing angles✓12 = ✓12 sin 1 23 = 22 ✓sin =131=† 0⇤ ⇤ [Lin ✓13 = 0. al.TBM
as sin2 3 , T = 3 † , ⇤0 , T23T ✓ 1,and sin2 T Shimizu et 11,Luhn,King 11,...]
✓
sin 10,
⌦e Me Me ⌦e = 0 e 1 Me Me ⇢ ⌦e = ⇢diag ⌦T Me Me ⌦⇤ ⇢⇤
⌦ ⇢ 0 2 e
†
e diag

6. mixing amongst all three mass eigenstates. Therefore this symmetry cannot b
2. Discrete Lagrangianbrokenit )has to be⇢(gMixingto be G⌫ mismatched horizontal sym- a
Lagrangian but mixing to beof the entireunderstood as a consequence broken subgroups G and
certain it has †patterns can be di↵erent subgroupshas to
to Lagrangian but it G and of (with
of the entireSymmetrye )⌦but⇢(geand, Leptonbroken e⇢(g⌫di↵erent to di↵erent subgroups Ge G
⌦e ⇢(g Groups diag but † has to=be broken to di↵erent subgroups G (6) e
of lepton and neutrino sectors,⌦respectively. If the[11–13;
= it ⌫ )⌦⌫ )
ection) in metries acting on the echarged lepton and⌫ neutrino sectorsdiag neutrino sectors, respectively. e
the chargedthe entire Lagrangian in the charged lepton and 1 26–28]4 . Let us assume If
Lepton mixing from discrete groups
1 trivial intersection)
0 fermions
trivial intersection) inthat there is as (discrete) symmetry group G under which therespectively. If
the charged the charged lepton and sectors, respectively. If the
transform a leptoncos ✓ neutrino sin ✓
and
0 where ⇢(g) thisare diagonal unitary matrices. These conditions idetermine ⌦ sectors, to a
1 0 purposeintersection) in
trivial
for
diag
0 e neutrino e , ⌦⌫ up left-handed
f
C transform as
transform A matrix K and 13 (✓, complete matrices P
1
0 B under a C
i as doublets L = (⌫, e) transformflavour groupfaithful unitary 3-dimensional representation
p
diagonal lepton
p
phase
e ! ⇢(ge )e,
and U permutation
⌫ e,⌫ ⇢(g⌫ )⌫,
!
)=@ 0 1
ge 2 Ge , g⌫! ⇢(ge,)e,
e 2 G⌫ e,⌫ 0 A. (13)
⌫ ! ⇢(g⌫ )⌫, (4) ge 2 Ge , g⌫ 2 G⌫ ,
Gf
2 0.70 2
1
0 0.652: G ! GL(3, ):
⇢i
e i sin
therefore restrict the ✓ 0 matrices have to fulfil
cos ✓
arged lepton fmasses and we the symmetry to hold, ourselves to the abelian case.
p p
2
metry to symmetry of (Me Me+) e ! ⇢(ge )e, fulfil⌦⌫ !)e, e,⌫ .)⌫, ! ⇢(g⌫ )⌫, Ge , ge residual ,⌫g⌫ 2 G⌫ ,
residual hold, the mass matrices
0.60
for have ⌦e,⌫ e ! e,⌫ Ke,⌫ P ⌫ mass of 13 ge 2
to ! ⇢(ge ⇢(g ⌫ q rotation 2 G , (7)
⌫ 2 Ge symmetry of M
ict ourselves to the case of Majorana ! ⇢(g)L, L neutrinos, which .implies that there
g2G f (3)
neutrino and charged mixing
lepton e )T Me Mand e )⇤
p masses † we
= M M † restrict ourselves M ⇢(g ) = M⌫ .
therefore
does not completely fixsymmetry to hold, thematrix yet, as the first fulfil third
the leptonic ⇢(g )T to the abelian c
to and
0.55 T
sin2 Hq23 L
⇢(g )T M Me ⇢(gefor the e M †
† LH⇢(g
leptons 3-dim e ⇢(g
)⇤ to matrices ⇢(g and ⌫mass rep. and
are 2
ambiguities of the last equation, V (5) given⌫by ⌦⌫ . ⌫
⇢(g ) M ⇢(g ) = M . e e
plex eeigenvalueWe = Mhold, the ourselves⌫ (i) they all matrices satisfy ⇢(g⌫ ) which (ii) there is th
therefore have are unequal and e,⌫
It follows efrom Eqn. (5) thate up to the ⌫ ) matrices ⌫have to fulfil e,⌫
of furtherdata clearly shows to the case lepton masses neutrinos, = 1, implies that
and
d=
for the symmetry correspondingmass 2
0.50
the samebe seen as
and the can
⌫
The experimental restrict eigenstatesthat beofrotated into each other
Majorana0.1 0.2
This can det ⇢(g⌫ ) = 1. By further erequiringbe a-0.2 distinguishable Majoranaa degenerate ma
-0.1 0.0
er choose be a non-abelian a Choosing eigenvalue to degenerate mass⌫group would therefore a symmetry)2 =
cannot be three would M Mto of three
0.45
ng the to mixing amongst allgroup⇢(gfix eigenstates.the = M Mthis symmetry cannot to satisfy = M .
or G⌫ symmetry. To T mass the G † a ⇤ matrices haveand they lead M
G or non-abelian )
complex )T lead ⌫ ⇢(g )Therefore †⇢(g spectrum, ⇢(g )T be ⇢(g ) ⇢(g⌫
oup G⌫ 0.35 restrictedwe canefurtherbutT three M⇤ ⌦Z)2=⇢TZBy T Mdi↵erent ⇤subgroups=Z2and2G⌫ .(with
is cannot⇢(gedecomposed⇢(g⇢ )M= † ⇢ e Mecannotone-dimensionalS,UthreeG⌫ )xZ M⌫ one-dim
completely representations e matrix weG = to into
e ⇤ e mixing and enlarge G ⌫
esentations e= entireMe 3e e
G the look at the resentations ofd=0e ⇢(gG .symmetry egenerator U withwith the case of three d
T⌦T =Z e↵ecteof addingorthe This scenario is not diag
M 1 e into it M to ⌫ †brokenbe decomposed )rep- ⇢(g e⌫ = ⌫
0.40
as † ⇢(g T three distinguishable 1
)Lagrangian⌦Tchoosehasinequivalent 2and ebe†requiring M⌫1 ⌫ inequivalent Major
e e ⌫
of and
be be Mtheir e
toM † ⌦⇤ = Klein group be=⇥1. ⌦ TBM M0⌦⇤ ⇢ ⌫to determine 0
the det ⇤ . to
To
further able
nerator. Let us 0 0 1 0 e e Ge e⌫ diag e e 1 compatible
e
trivial intersection) GgroupG⌫⌫ to restricted to be the Klein group Z2 0lead. To is degenerate
ionse of rows and columns) 4 or
of G or G⌫ Choosing
scenario inthe charged lepton and neutrino sectors, 0
G is matrix from the distinguished respectively. be 1 0 0
the e compatible with non-abelian group would ⇥ Z2 it a able to determ
0.30 . This neutrinosis not the mixingbe a the case of threegroup structure to If the fermions m
@ 0 in [26; 27]. A ⇢(U ) = @ 0 0 1 A
⇢(T e or asto0 todiagonal0 of rows and group and convention matrix0from the phase mass sp
0.05 ) = @ A 1
presentation the =
⇢(S) 1
Choosingdiagonal (onlypermutations27].followisthesingletswhen conjugated by three inequivalent one-dim
has topresentationtransform be inequivalentinvariant decomposed sameais true for
betransform 0 their 1 asinWe non-abelian columns) would lead to a degenerate structure i
G (up convention a here
G⌫ a representations cannot be of G .mixing into arbitrary group
matrix
e all neutrinos and
llow the as [26; 1 0 0 The0 0 1 0 1 0
1 0 and permutation freedom can be ⌫
necessary 0 cannotneutrinos transform as inequivalentinequivalent The same is true
phasing to have all be decomposed into three singlets of G⌫ . one-dimensio
Bbe smaller than Z . We can now determine case
C
matrix) and the resentations of G or G . This scenario is not to bring it into the form of three
used compatible with the
onstheir representations 3 ⌘! ⇢(ge0 0 1 ! ⇢(g⌫ )⌫, 3 ge (Z xZ emost 2 G⌫ , choice if mixing angles do not
as which shows⇢(U ) = U cannote )e, ⌫ ⌫ A .
0.04
that Ge
2 , m2 , m but and analogously@ 22 smaller than (14)
2
the charged e which shows that G cannot be G2 , g⌫ considerations we
(Z3 smallest choice, 2 ), can also be leptons for ⌦⌫ . From thesee group theoreticalgeneral Z3 . We can now determ
diag(me µ ⌧
(4)
he continuous) matricesWe ehere.⌫ This misaligned non-commuting ⌫ that satisfy on27]. the Majorana three distin
resentations of thee4mixing⌦ follow the0 1matricesand ,conventiondependwith & case of s)
G or G⌫via thatunitary 0 is not compatible masses
2 scenario
sin2 Hq13 L
0.03
unitary ⌦ , the satisfy
can thus determine the PMNS matrix presentation ⌦e ⌦
for the symmetry to hold, the mass matrices have to fulfil
in [26;
4 0.02 symmetries lead to
We †here follow the† presentation and convention in [26; 27]. †
✓ ⌦†e )⇤ )⌦ = e ⌦⌫ diag ,
⌦† ⇢(g M 1,⇢(g ⌫ )
value⌦e0.01 to zero, ⇢(g⇢(gdiagMe Me ⇢(gU⇢(ge⌫)⌦=⌫⌦= ⇢(gand thus ⌫ ⇢(g⌫mixing) matrix (8)
of⇢(ge )⌦e = ⌦U ⇢(U )⌦U =† diag( =1, ee M†1),e )anddiag ⇢(g⌫ )T M⌫⌫⇢(g⌫⇢(g⌫ )diag
e ) e )T , ⌫e PMNS † ⌦ the )⌦ = = M (6)
2 ⌫. (5)
mixing matrix determined from symmetry
tri-bimaximal mixing ⇢(g)diag are diagonal unitary matrices. These conditions determine ⌦e , ⌦⌫ up t
where (TBM) be form up to interchanging of rows/columns and
re [He, Keum,Choosing0.30 eofmatrices.xZ0.34non-abelian group2would lead to that it 0.65 not possible
up to07, 0.00Volkas 06; G G rows and columns. It should not determine 0.55a ,degenerate mass spectrum,
as and permutation matrix,q and permutation tosin Hq23 L lepton one-dimensional
sin Hq12 L
diagonal unitary or= G⌫ to=Z2 2a
Lam
a 08;
permutation
0.28 0.32
S,U These conditions 0.40 0.45 0.50 ⌦e 0.60 ⌫ is 0.70 a
0.36 0.30 0.35 be surprising diagonal phase to
⌦ up matrix
2
0 as 1 predict
to uniquely pin diagonal phase matrix Kbe decomposed into matrices Pe,⌫
Altarelli,Feruglio 05, representations cannot e,⌫ it is not
2
matrix Ke,⌫ theirdown the mixing matrices 2Pe,⌫p possible three inequivalentmasses in this rep-
Feruglio,
1
0
approach. resentations of Ge or G⌫ . This scenario is3not compatible with the case of three distinguished
B 3 C G =S
Deviations = from tri-bimaximal mixing of the form! ⌦e,⌫UHPPe,⌫13=Z2 ) (2.29). The yellow poi
1:Hagedorn,Toroop 11] ⌦† ⌦ = U B p 1 1 ⌦e,⌫ U C , Ke,⌫ S U . (✓,
1 =
PMNS here to U HPS ⌘ @ and to p
Letthe now try efollow the this e,⌫ Ke,⌫ Pe,⌫ . some interesting cases. We have0 (15) codei in the to
Uus 4 apply! ⌦ machinery convention inp2 27]. A 1
TBM, We ⌦e,⌫ give the deviations 3
continuous lines presentation 6 with [26; angle ✓ given ⌦ the colour the e sin ✓
the by seen that 0
(7)
cos↵✓
smallest n ⇡ forsymmetry .in5,Eqn. (5) n p up to the ambiguitiesparabola T |T@ = E V⇠ are given A
follows . , where that lepton1sector p given UT the = BM 3
1 1
er for =residual Itn = 0, . fromthe charged=60 ispthe outermostbyofMGe = UTetc. The0one,Ztwo andby ⌦
is a M last equation, = 13 . 0
e,⌫ thr
tri-bimaximal
5 2 3 2 i
e sin ✓ 0 cos ✓
qn. (5)athat upwhere can [39] are1indicated by 1 dotted, dashede,⌫ are given by contours, respective
ons of a basis global the be seen as is given by equation, V and continuous ⌦e,⌫ .
We use recent This fit generator of the last
to the ambiguities 2 2
mixing (TBM) sin2 ✓12 = mixing angles in the2 direction of[Lin 10,experimental (TM2) ✓ ⇠ .1
, sin 1 23 = 2 sin ✓131= 0
✓ 0 , tri-maximal mixing
onds to the mixingcan shift sin2 ✓12 = 3 † , ⇤ 2 ✓23T = 1,and ⇤ the13 = 0. al.TBM
rn of perturbations angles the 3 T
as sin T sin2 ✓ Shimizu et 11,Luhn,King 11,...]
data for
⌦e Me Me ⌦e = 0 e 1 Me Me ⇢⇤ ⌦e = ⇢T ⌦T Me Me ⌦⇤ ⇢⇤
⌦ ⇢ 0 2 † diag e
†
e diag

7. transform as
certain mixing patterns can be understood as a consequence of mismatched horizontal sym-
0.55
metries acting on the charged lepton and neutrino sectors [11–13; 26–28]4 . Let us assume
0.50
Lepton mixing from discrete groups
e ! ⇢(ge )e,
Ê
⌫ ! ⇢(g⌫ )⌫, g e 2 Ge , g ⌫ 2 G⌫ ,
for this purpose that there is a (discrete) symmetry group Gf under which the left-handed
for the complete flavour group
lepton doublets L = (⌫, e) 0.2 symmetryunder a faithful unitary 3-dimensional representation
transform to hold, the mass matrices have to fulfil
residual symmetry of (Me Me+) 0.0
-0.4 -0.2 0.4
residual symmetry of M
0.45 ⇢ : Gf ! GL(3, ):q T † ⇤ † T G f
⇢(ge ) Me Me ⇢(ge ) = Me Me and ⇢(g⌫ ) M⌫ ⇢(g⌫ ) = M⌫ .
0.40Ge= T =Z3
0 1 f
Choosing Ge or G⌫ to be a non-abelian group would lead to a degenerate ma
L ! ⇢(g)L, g2G . (3)
0 1 0 LH leptons 3-dim rep.
n 2 2
asclearlyrepresentations cannot be decomposed into three inequivalent one-dim
their shows (i) that all lepton masses are unequal and (ii) there is G = S,U =Z xZ
⇢(T ) = @ 0 0 1 Adata
0.35 The experimental 0 1 0 1
resentations of Ge or G⌫ . This scenario is not compatible with the 1 0of three d
case
mixing 1 0 0 all three mass eigenstates. Therefore this
amongst 1symmetry cannot be a symmetry 0
0 0
4 ⇢(S) = @ 0 1 0 A ⇢(Un ) = @ 0 0 zn A
0.30
0.400.30
0.40 of the0.40 0.45 Lagrangian We here follow to be broken and di↵erent subgroups Ge and G⌫ (with
0.35 entire 0.50 0.60 has the presentation to convention in [26; 27].
0.55 but it 0.65 0.70 ⇤
0.05
0.05 0.05 0 0 1 0 zn 0
trivial intersection) in the charged lepton and neutrino sectors, respectively. If the fermions
Ê
4transform as
Ê
Ê
2 hzn i ⇠ Zn
Ê
=
0.04
0.04
• e 2 Ge , over all ,discrete groups of (4)
g Scan g⌫ 2 G⌫
e ! ⇢(ge )e, size 0.04
⌫ ! ⇢(g⌫ )⌫,
smaller than 1556 with Ge=Z3,
for Ê symmetry to hold, the mass matrices have to fulfil =Z xZ
7 the
Ê
Ê G Ê
0.03
0.03 0.03
2 2
5 Ê Ê ⇤ • all give a TM2-like correction
sin 2Hq13L
Ê Ê
T † † T
⇢(ge ) Me Me ⇢(ge ) = Me Me 1 and ⇢(g⌫ ) M⌫ ⇢(g⌫ ) = M⌫ .
0 (5)
Ê Ê
Ê
Ê
16 cos ✓ 0 sin ✓
Choosing Ge or G⌫ to be aÊnon-abelian group would lead to a degenerate mass spectrum, A
0.02
0.02
9
Ê Ê 0.02
U = UT BM @
Ê
0 1 0
as their representations cannot be decomposed into three inequivalent one-dimensional rep-
7 Ê
Ê
Ê Ê Ê sin ✓ 0 cos ✓
resentations of Ge or G⌫ . This scenario is not compatible with the case of three distinguished
8 Ê Ê
Ê Ê 1
with ✓ = arg(zn )
0.01
0.01 4 0.01
We here follow the presentation and convention in [26; 27].
5 Ê Ê Ê Ê 2
• vanishing CP phase !CP
7 Ê Ê
Ê Ê
Ê
Ê
Ê
Ê
• recent global fits indicate large deviation
16 2
sin 2Hq 23L
0.00
0.00 ÊÊ Ê 0.00
0.400.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70
from maximal atmospheric mixing
[MH, K.S. Lim, M. Lindner 1212.2411(PLB)]

8. transform as
certain mixing patterns can be understood as a consequence of mismatched horizontal sym-
Ê Ê
0.55 0.55
Ê
Ê
metries acting on the charged lepton and neutrino sectors [11–13; 26–28]4 . Let us assume
Ê
Lepton mixing from discrete groups
Ê
sin 2Hq 23L e ! ⇢(ge )e, ⌫ ! ⇢(g⌫ )⌫, g e 2 Ge , g ⌫ 2 G⌫ ,
for this purpose that there is a (discrete) symmetry group Gf under which the left-handed
0.50 Ê 0.50 Ê
complete flavour group -0.4
for the symmetry to hold, the mass matrices have to fulfil
lepton doublets L = (⌫, e) transform under a faithful unitary 3-dimensional representation
-0.2
residual symmetry of (Me Me+) 16 Ê 7
0.0 0.2 0.4
residual symmetry of M
Gf
Ê Ê
Ê
⇢ : Gf ! GL(3, ):
0.45
Ê
Ê 5
8
Ê
Ê
T
0.45
†
⇢(ge ) Me Me ⇢(ge ) = Me Me ⇤
and † q T
⇢(g⌫ ) M⌫ ⇢(g⌫ ) = M⌫ .
Ê7
G = T =Z
Ê
e
0.40 Ê 9
3 1 Choosing Ge L ! ⇢(g)L, a non-abelian group would lead to a degenerate ma
0.40 Ê
g 2 Gf . (3)
0 16 Ê Ê
or G⌫ to be Ê
Ê 7
G = S,U =Z xZ
5 Ê
0 1 0 Ê LH leptons 3-dim rep.
2 2
Ê
n
asclearlyrepresentations0.35all leptondecomposed into three inequivalent one-dim
theirÊ shows (i) thatcannot be masses are unequal and (ii) there is
⇢(T ) = @ 0 0 1 Adata
The experimental
0.35 4 0 1 0 1
Ê
resentations of Ge or G . This scenario is not compatible with the 1 0of three d
case 0
mixing 1 0 0 all three mass eigenstates. ⌫Therefore this1symmetry cannot be a symmetry
amongst 0 0
0.30 4 0.30 ⇢(S) = @ 0 1 0 A ⇢(Un ) = @ 0 0 z A
of the entire Lagrangian We here0.36 0.38 be 0.400.30 0.35 di↵erent subgroups Ge and0.65 ⌫ 0.70⇤ n
0.26
0.26 0.28
0.28 0.30
0.30 0.32
0.32 but it0.36 0.38 presentation and convention in 0.50 27].
0.34
0.34 follow to
has the 0.40
broken to 0.40 0.45 [26; 0.55 0.60 G (with
0.05 0.05
0.05
0 0 1 0 zn 0
0.05
trivial intersection) in the charged lepton and neutrino sectors, respectively. If the fermions
transform as 4 Ê 4 Ê
Ê 2 hzÊ i ⇠ Zn
Ê
n = Ê
• e 2 Ge , over all ,discrete groups of (4)
0.04 0.04
0.04 0.04
e ! ⇢(ge )e, g Scan g⌫ 2 G⌫⌫ ! ⇢(g⌫ )⌫, size
smaller than 1556 with Ge=Z3,
Ê
Ê 7mass matrices have to Ê
Ê
Ê Ê Ê
for the symmetry to hold, the
0.03 0.03
0.03
7 fulfil
G =Z2xZ2 0.03
Ê 5 5 Ê Ê
sin 2Hq13L
sin 2Hq13L
• all give a TM2-like correction
Ê Ê Ê
T † ⇤ † T
⇢(ge ) Me MeÊ e ) = Me Me
16 ⇢(g Ê and Ê⇢(g⌫ ) M⌫ ⇢(g⌫ ) = M⌫ . Ê
Ê
1 0 Ê (5)
16
cos ✓ 0 sin ✓
Choosing Ge or G⌫ to be Ê 9non-abelian group would 9lead to a degenerate mass spectrum, A
a Ê Ê
U = UT BM @
0.02 Ê 0.02
0.02 0.02 Ê Ê
0 1 0
as their representations cannot be decomposed into threeÊinequivalent one-dimensional rep-
Ê7 Ê Ê
resentations of Ge or G⌫8 Ê
Ê
7 sin ✓ 0 cos ✓
. This scenario is not compatible with the case of three distinguished
Ê Ê
8 Ê 1 Ê
Ê Ê Ê
0.01
4
0.01
0.01
with ✓ = arg(zn ) 0.01
Ê 5
We here follow the presentation and convention in [26; 27].
Ê
5 Ê 2 Ê Ê Ê
16 Ê
Ê 7
Ê
Ê
• vanishing CP phase !CP
7 ÊÊ ÊÊ Ê
Ê Ê
Ê
2 • recent global fits indicate 0.65 0.70
16
large deviation
sin 2Hq12L from maximal L
sin 2Hq 23 atmospheric mixing
0.00
0.26 0.28
Ê
0.30 0.32
Ê0.34 0.36 0.38
0.00
0.00
0.400.30 0.35 0.40 0.45
Ê
0.50 0.55 0.60
Ê 0.00
Ê
[MH, K.S. Lim, M. Lindner 1212.2411(PLB)]

9. transform as
certain mixing patterns can be understood as a consequence of mismatched horizontal sym-
metries acting on the charged lepton and neutrino sectors [11–13; 26–28]4 . Let us assume
Lepton mixing from discrete groupse ! ⇢(ge )e, ⌫ ! ⇢(g⌫ )⌫, g e 2 Ge , g ⌫ 2 G⌫ ,
for this purpose that there is a (discrete) symmetry group Gf under which the left-handed
complete flavour group
for the symmetry to hold, the mass matrices have to fulfil
lepton doublets L = (⌫, e) transform under a faithful unitary 3-dimensional representation
residual symmetry of (Me Me+) residual symmetry of M
⇢ : Gf ! GL(3, ): T † ⇤ G†
f T
⇢(ge ) Me Me ⇢(ge ) = Me Me and ⇢(g⌫ ) M⌫ ⇢(g⌫ ) = M⌫ .
Ge= T =Z3
0 1 f L ! ⇢(g)L, g2G .
Choosing Ge or G⌫ to be a non-abelian group would lead to a degenerate ma
(3)
0 1 0 LH leptons 3-dim rep.
n 2 2 G = S,U
asclearlyrepresentations cannot be decomposed into three inequivalent one-dim
their shows (i) that all lepton masses are unequal and (ii) there is =Z xZ
⇢(T ) = @ 0 0 1 Adata
The experimental 0 1 0 1
resentations of Ge or G⌫ . This scenario is not compatible with the 1 0of three d
case
mixing 1 0 0 all three mass eigenstates. Therefore this
amongst 1symmetry cannot be a symmetry 0
0 0
4 ⇢(S) = @ 0 1 0 A ⇢(Un ) = @ 0 0 zn A
of the entire Lagrangian We here follow to be broken and di↵erent subgroups Ge and G⌫ (with
but it has the presentation to convention in [26; 27].
trivial intersection) sin 2the Lcharged lepton and neutrino sectors, respectively. If the fermions
in Hq
⇤
0 0 1 0 zn 0
transform as
0.26 0.28 0.30 0.32
12
0.34 0.36 0.38 0.40
2 hzn i ⇠ Zn
=
0.70 0.70
0.65 e ! ⇢(gÊ)e,
e Ê
• e 2 Ge , over all ,discrete groups of (4)
g Scan g⌫ 2 G⌫
⌫ ! ⇢(g⌫ )⌫,
0.65
size
Ê
Ê Ê
Ê smaller than 1556 with Ge=Z3,
Ê mass matrices have to fulfil
for the symmetry to hold, Ê
the G =Z2xZ2
Ê
0.60 0.60Ê
Ê Ê
T
Ê
ʆ ⇤ †
Ê
• all give a TM2-like correction
T
0.55 ⇢(ge ) Me Me ⇢(ge ) = Me Me 0.55
Ê
Ê
and ⇢(g⌫ ) M⌫ ⇢(g⌫ ) = M⌫ .
0 1 (5)
Ê Ê
Ê
cos ✓ 0 sin ✓
sin 2Hq 23L
Choosing Ge or G⌫ to be a non-abelian group would lead to a degenerate mass spectrum, A
0.50 Ê 0.50
U = UT BM @ 0 1 0 Ê
as their representations cannot be decomposed into three-0.2
16 Ê 7
-0.4 inequivalent one-dimensional rep-
0.0 0.2 0.4
Ê sin ✓ 0 cos ✓
Ê
Ê
resentations of Ge or G⌫ . Ê 5 scenario is not compatible with the q
0.45 0.45
This
Ê
case of three distinguished
8 Ê
Ê7
Ê
Ê
1
with ✓ = arg(zn )
4
Ê 9
0.40 We here follow the presentation and convention in [26; 27].
Ê 0.40
Ê16 Ê
Ê 5
Ê 2
• vanishing CP phase !CP
Ê
Ê7
Ê
4 Ê
• recent global fits indicate large deviation
0.35 Ê 0.35
2
0.30 0.30 from maximal atmospheric mixing
0.05
0.26
0.26 0.28
0.28 0.30
0.30 0.32
0.32 0.34
0.34 0.36
0.36 0.38
0.38 0.400.30
0.40
0.05
0.05 [MH, 0.40 Lim, M. Lindner0.60 0.65 0.70
0.35
K.S. 0.45 0.50 0.55 1212.2411(PLB)]
0.05

10. Lepton mixing from discrete groups
n G GAP-Id sin2 (✓12 ) sin2 (✓13 ) sin2 (✓23 )
sin 2Hq12L 5 (6 · 102 ) [600, 179] 0.3432 0.0288 0.3791
0.70
0.26 0.28 0.30 0.32 0.34 0.36 0.38 0.40
0.70 0.3432 0.0288 0.6209
9 (Z18 ⇥ Z6 ) o S3 [648, 259] 0.3402 0.0201 0.3992
0.65 Ê Ê 0.65
0.3402 0.0201 0.6008
Ê
Ê
Ê
(6 · 162 )
Ê
Ê Ê
16 n.a. 0.3420 0.0254 0.3867
0.60 Ê Ê 0.60
Ê Ê
0.3420 0.0254 0.6134
Ê Ê
0.55
Ê
Ê
0.55
Ê
Ê
Ê
Ê
sin 2Hq 23L
Table 3: Mixing angles which are compatible with experimental results generated by flavor groups u
0.50 Ê 1536. The group identification function in SmallGroups is not available for group with order 1536.
0.50 Ê
-0.4 -0.2 0.0 0.2 0.4
16Ê 7
Ê Ê
Ê
always a Klein group,q 3 , U3 (n)i ⇠ Z2 ⇥ Z2 . For the group generated by T3 , S3 and U
0.45 0.45
Ê 5
Ê
8 Ê Ê hS =
Ê7 Ê
0.40 Ê 9 Ê
be finite, z has to be of the form given in Eqn. (20), as may be seen by looking at th
0.40
Ê 16 Ê
∆(6·4 )= T,S,U 〉
Ê
Ê 57 Ê
Ê Ê
2
element (U3 (n)T3 )2 = diag(z, z, z ⇤ 2 ) which is of finite order n 2 i↵ z n = 1. The requ
0.35 4 Ê Ê 0.35
4
⇠ Zn further fixes n to be the smallest n for which z n = 1. Note that di↵erent n
hzi =
∆(6·102)= T,S,U5〉 0.30
0.05
0.26
0.26 0.28
0.28 0.30
0.30 0.32
0.32 0.34
0.34 0.36
0.36 0.38
0.38
0.30
0.400.30
0.40
0.05
0.05
will 0.40 0.45 lead to 0.60 0.65 0.70
0.05
[Torop, Hagedorn,
of z 0.35 in general 0.50 0.55 di↵erent leptonic mixing angles via Eq. (19), as can be
Figure 2. The group generated by T3 , S3 and U3 (n) is always the same due to the requ
4 Ê
Ê
=
4 Ê
Ê
Feruglio 11, King, Luhn,
hzi ⇠ÊZn . The names of the groups generated for n = 4, . . . , 16 can be found in Tab Ê
0.04 the groups (96) (n = 4) and (384) (n = 8) have been obtainedDing 12]
0.04
0.04 0.04
Steward 12, before in [27]. N
not all of the groups generated in this way can be classified as (6 · n2 ), e.g. th
∆(6·16 )= T,S,U 〉
Ê
Ê
Ê 7 (Z18 ⇥ Ê 6 ) o S3 5 . Another surprising observation is that all the2groups which are re
7 Z
Ê
Ê Ê
0.03
Ê
0.03
0.03
a 0.03
to thoseÊshown in Figure 2 giveÊ prediction of CP = ⇡ or CP = 0. In general other 16
sin 2Hq13L
sin 2Hq13L
Ê 5 5 Ê Ê
16 Ê Ê
that we16have scanned which lie Ê
Ê Ê
outside the region shown in Figure 2 do give non-trivi
Ê
0.02 Ê Ê 9 CP phases. The predictions for mixing angles for all groups of size smaller or equal th
0.02
0.02 Ê Ê
Ê 0.02 Ê
9
Ê Ê 7
groups is presented in Figure 2. It should be clear that if one allows for groups of a
7 Ê
Ê
Ê 18 Ê
6 Ê
size, the parabola depicted in Figure 2 will be densely covered. As one can see, the
3 (Z ⇥ Z ) o S
8 Ê Ê
Ê Ê Ê Ê
0 0.01
1 Ê 5
0.01
0.01 8
patterns corresponding to n = 5, n = 9 0.01 n = 16 give a good descriptions of the
5 Ê Ê
and
3 3 9
⇠ hS , T , U i
=
1 0 0
Ê Ê Ê
Ê
mixing matrix. Ê Table 3 for the resulting mixing angles.
See Ê
16 Ê 7
Ê Ê Ê
7 Ê Ê
⇢(Un ) = @ 0 0 zn A
Ê Ê Ê
16
sin Hq12L G⌫ = Z2 ⇥ Z2 ,23L e | > 3
sin 2Hq |G
0.00
0.26 0.28
Ê
0.30 0.32
Ê0.34 0.36 0.38
0.00
0.00
0.400.30 0.35 0.40 0.45
Ê
Ê
0.50
Ê
0.55 0.60 0.65
0.00
0.70
⇤ 2 3.2
0 zn 0
hzn i ⇠ Zn The leptonic mixing angles (black circles) determined from[MH,chargedLim, M.1536 are We1212.2411(PLB)]
In this section we will discuss the result of all the neutrino mixing angles scanned by
= 2:
Figure
the condition on the K.S. lepton subgroups.
our group scan up to order
Lindner allow G to be any abelian group