Lepton Flavor Violation in Complex SUSY Seesaw Models with Nearly Tribimaximal Mixing

3.6 MB·English

by  Frank F. Deppisch

MAN/HEP/2010/20
SISSA 41/2010/EP
Lepton Flavor Violation in Complex SUSY
Seesaw Models with Nearly Tribimaximal Mixing
1
1
0
2 Frank F. Deppischa†, Florian Plentingerb‡ and Gerhart Seidlc§

n
a
J

2 aSchool of Physics and Astronomy, University of Manchester

Manchester M13 9PL, United Kingdom
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p bSISSA and INFN-Sezione di Trieste
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p via Bonomea 265, 34136 Trieste, Italy
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cInstitut fu¨r Theoretische Physik und Astrophysik
[
Universit¨at Wu¨rzburg, 97074 Wu¨rzburg, Germany
2
v
4
0
4
1
1. Abstract
1
0 We survey the lepton flavor violation branching ratios Br(µ → eγ), Br(τ → µγ), and
1 Br(τ → eγ) in mSUGRA for a broad class of lepton mass matrix textures that give nearly
:
v tribimaximalleptonmixing. Smallneutrinomassesaregeneratedbythetype-Iseesawmech-
i
X anismwithnon-degenerateright-handedneutrinomasses. Thetexturesexhibitahierarchical
r mass pattern and can be understood from flavor models giving rise to large leptonic mixing.
a
We study the branching ratios for the most general CP-violating forms of the textures. It
is demonstrated that the branching ratios can be enhanced by 2-3 orders of magnitude as
compared to the CP-conserving case. The branching ratios exhibit, however, a strong depen-
dence on the choice of the phases in the Lagrangian which affects the significance of flavor
models. In particular, for general CP-phases, the lepton flavor violating rates appear to be
essentially uncorrelated with the possible high- and low-energy lepton mixing parameters,
such as the reactor angle.
†E-mail: [email protected]
‡E-mail: [email protected]
§E-mail: [email protected]
1 Introduction
Neutrinooscillationexperimentshaveduringthepastdecadepinneddowntheneutrinomass
and mixing parameters to a remarkable precision [1, 2, 3, 4]. Global fits [5] tell us that the
solar and atmospheric neutrino mass squared differences are (at 1σ)
∆m2 = 7.65+0.23 ×10−5eV2, |∆m2 | = 2.40+0.12 ×10−3eV2, (1)
(cid:12) −0.20 atm −0.11
whereas the solar, atmospheric and reactor1 mixing angle are respectively given by
θ = (32.6+3.2)◦, θ = (45.0+4.1 )◦, θ ≤ 13.2◦ (3σ). (2)
12 −2.7 23 −10.3 13
The leptonic Pontecorvo-Maki-Nakagawa-Sakata (PMNS) mixing matrix U [6] can be
PMNS
well approximated by the Harrison-Perkins-Scott (HPS) tribimaximal mixing matrix U
HPS
[7] (up to phases) as
(cid:113) 
2 √1 0
3 3
UPMNS ≈ UHPS = −√1 √1 √1 . (3)
6 3 2
√1 −√1 √1
6 3 2
In U , the solar and the atmospheric angle are given by θ ≈ 35◦ and θ = 45◦, whereas
HPS 12 23
the reactor angle θ vanishes. The measured PMNS mixing angles may thus be treated
13
as deviations from exact tribimaximal mixing [8, 9], describing nearly tribimaximal lepton
mixing [10].
The smallness of the absolute neutrino mass scale m (cid:39) 5×10−2eV can be understood
ν
in terms of seesaw mechanism [11, 12], which establishes a connection between low-energy
neutrino observables and physics near the unification scale M ≈ 2.5 × 1016GeV [13]. In
X
such a context, neutrino oscillations can therefore probe high-scale theories of flavor.
Itisinterestingtoaskwhethertherearepossibilitiestotestmassandleptonflavormodels
in other than only neutrino oscillation experiments. Such a possibility is offered by charged
lepton flavor violation (LFV) in supersymmetry (SUSY). In the standard model (SM), LFV
is absent and adding right-handed neutrinos to the SM only tiny LFV effects that are sup-
pressed by the smallness of the neutrino masses. In SUSY, however, virtual effects of the
right-handed neutrinos and their superpartners affect the renormalization group equations
(RGE) of the slepton masses and trilinear couplings and induce observable branching ratios
(BRs) of LFV decays [14].
One attractive feature of the seesaw mechanism is that it allows for leptogenesis via
the decay of the right-handed neutrinos [15] (for reviews see [16]). Barring special cases,
however, the complex Yukawa couplings providing the necessary CP-violating phases can
generally lead to a large enhancement of the LFV rates by large factors (cid:38) 104 [17]. If these
CP phases are not fixed, their presence may obscure possible predictions from flavor models
and, furthermore, make it difficult to distinguish experimentally different candidate theories
of flavor. For a discussion of LFV in certain quark-lepton complementarity scenarios see for
example [18].
1Note, the current best-fit value for the reactor angle is θ =6.5◦.
13
2
In this paper, we study the LFV decays µ → eγ, τ → µγ, and τ → eγ, in minimal
supergravity (mSUGRA) for a large class of CP-conserving and CP-violating lepton mass
matrix textures that yield nearly tribimaximal lepton mixing. The textures are formulated
at the level of the Lagrangian and are characterized by hierarchical entries that can be
generated in flavor models using, e.g. the Froggatt-Nielsen (FN) mechanism [19]. The tex-
tures implement the idea of quark-lepton complementarity (QLC) [20] in a general way:
They realize nearly tribimaximal lepton mixing by taking large contributions from differ-
ent lepton sectors, i.e. the charged lepton and the left- and right-handed neutrino sector,
into account. The FN mechanism has, for instance, been used in Refs. [21, 22, 23, 25] to
construct charged lepton and even neutrino mass textures which can be implemented by
discrete flavor symmetries [23, 24, 26]. Many models have been proposed in the literature
to reproduce tribimaximal leptonic mixing using non-Abelian flavor symmetries (for early
models based on A (cid:39) Z (cid:110)(Z ×Z ) and examples using the double covering group of A ,
4 3 2 2 4
see Refs. [27] and [28]). We demonstrate that the inclusion of random CP-violating phases
at the level of the Lagrangian can enhance the branching ratios typically up to 2-3 orders
of magnitude. Moreover, we find a strong dependence of the LFV-rates on the choice of
the phases in the Lagrangian. As a consequence, it appears that the assignment of random
CP-violating phases in the Lagrangian in general erase possible correlations between LFV
rates and the PMNS mixing parameters of the low scale theory.
The paper is organized as follows: In Sec. 2, we introduce the SUSY seesaw mechanism
and its different parametrizations. Charged lepton flavor violation in SUSY is described in
Sec. 3. The types of textures for which we study LFV are introduced in Sec. 4, and our
results for LFV are then presented in Sec. 5. Finally, in Sec. 6, we give our summary and
conclusions.
2 SUSY Seesaw Mechanism
In what follows, we assume that the left-handed active neutrinos acquire their masses via the
type-I seesaw mechanism [11]. We consider the seesaw mechanism in SUSY, which differs
from the usual seesaw mechanism in the respect that it involves two Higgs doublets. In the
type-I seesaw mechanism, the part of the superpotential generating the charged lepton and
neutrino masses is given by
1
W = −(Y ) ec(cid:96) H −(Y ) νc(cid:96) H + (M ) νcνc, (4)
(cid:96) ij i j 1 D ij i j 2 2 R ij i j
where(cid:96) = (ν ,e )T,ec,andνc (i = 1,2,3isthegenerationindex)arethemattersuperfieldsof
i i i i i
the left-handed lepton doublets, right-handed charged leptons, and right-handed SM singlet
neutrinos, respectively. In (4), H and H are the usual Higgs superfield doublets generating
1 2
the down- and up-type masses, respectively, and Y and Y are the 3 × 3 Dirac Yukawa
(cid:96) D
coupling matrices of the charged leptons and neutrinos, whereas M is the 3×3 Majorana
R
mass matrix of the right-handed neutrinos. After electroweak symmetry breaking, the Higgs
doublets develop vacuum expectation values (cid:104)H0(cid:105), where (cid:104)H0(cid:105) = vsinβ with v = 174GeV
i 2
3
and tanβ = (cid:104)H0(cid:105)/(cid:104)H0(cid:105). The resulting lepton mass terms become
2 1
1
L = −(M ) ece −(M ) νcν + (M ) νcνc +h.c., (5)
mass (cid:96) ij i j D ij i j 2 R ij i j
where M = (cid:104)H0(cid:105)Y is the charged lepton and M = (cid:104)H0(cid:105)Y the Dirac neutrino mass
(cid:96) 1 (cid:96) D 2 D
matrix. M and M are complex 3×3 matrices. The matrix M is complex and symmetric
(cid:96) D R
and has matrix elements of the order of the B −L breaking scale M ∼ 1014 GeV. After
B−L
integrating out the heavy right-handed neutrinos, we obtain the effective low-energy 3×3
neutrino Majorana mass matrix
M = MTM−1M = YTM−1Y (vsinβ)2, (6)
eff D R D D R D
giving masses ∼ 10−2 eV to the light neutrinos in agreement with observation. The seesaw
mechanism is attractive since it establishes a connection between the absolute neutrino mass
scale and M , which is close to the grand unified theory (GUT) scale M ∼ 1016GeV.
B−L X
The mass terms in (5,6) are diagonalized by unitary 3×3 matrices U which in their most
x
general form can be written as (cf. Refs. [21, 22])
U = D ·U(cid:98) ·K , x = (cid:96),(cid:96)(cid:48),D,D(cid:48),R,ν, (7a)
x x x x
where D and K are given by D = diag(eiϕ1,eiϕ2,eiϕ3) and K = diag(eiα1,eiα2,1) with
x x x x
ϕ ∈ [0,2π[ and α ∈ [0,π[ and
i i
 
c c s c s e−iδ(cid:98)
12 13 12 13 13
U(cid:98) =  −s c −c s s eiδ(cid:98) c c −s s s eiδ(cid:98) s c  (7b)
x  12 23 12 23 13 12 23 12 23 13 23 13 
s s −c c s eiδ(cid:98) −c s −s c s eiδ(cid:98) c c
12 23 12 23 13 12 23 12 23 13 23 13
ˆ ˆ
is a unitary matrix in the standard parametrization with s = sinθ , c = cosθ , where
ij ij ij ij
θˆ ∈ {θˆ ,θˆ ,θˆ } ∈ (cid:2)0, π(cid:3), and δ(cid:98)∈ [0,2π[. The mass matrices then read
ij 12 13 23 2
M = U∗MdiagUT, M = U∗ MdiagUT, M = U MdiagUT, M = U MdiagUT, (8)
(cid:96) (cid:96)(cid:48) (cid:96) (cid:96) D D(cid:48) D D R R R R eff ν eff ν
where the diagonal mass matrices
Mdiag = diag(m ,m ,m ), Mdiag = diag(mD,mD,mD), (9a)
(cid:96) e µ τ D 1 2 3
Mdiag = diag(mR,mR,mR), Mdiag = diag(m ,m ,m ), (9b)
R 1 2 3 eff 1 2 3
having positive mass eigenvalues. The PMNS mixing matrix is given by
U = U†U = U(cid:98) K , (10)
PMNS (cid:96) ν PMNS Maj
where the matrix U(cid:98) in (10) is described in the standard parametrization (7b) by the
PMNS
ˆ ˆ ˆ
solar angle θ = θ , the reactor angle θ = θ , the atmospheric angle θ = θ and the
12 12 13 13 23 23
Dirac CP-phase δˆ = δ. K = diag(eiφ1,eiφ2,1) contains the Majorana phases φ and φ
Maj 1 2
with φ ∈ [0,π[.2
1,2
2The phase matrix K has been absorbed into the right-handed charged lepton sector.
(cid:96)
4
Rotating the lepton doublets (cid:96) by U to the basis where M is diagonal, M is diago-
i (cid:96) (cid:96) eff
nalized by the PMNS matrix
U† M U∗ = Mdiag. (11)
PMNS eff PMNS eff
Unless stated otherwise, we will from now on work in the basis where the charged lepton and
heavy Majorana neutrino mass matrix are diagonal, i.e. where M = Mdiag and M = Mdiag.
(cid:96) (cid:96) R R
The significance of the mixing matrices U in (8) is that the mass terms in (5) may
x
be predicted by some flavor model such as the Froggatt-Nielsen mechanism [19] or discrete
flavor symmetries [23, 24, 26]. Therefore, by tracing U back to the matrices U , which
PMNS x
describe the high-energy lepton mixing, one can gain better understanding of the observable
PMNS parameters in terms of a possible fundamental theory of flavor.
Flavor models do not fix the absolute scale of the Yukawa coupling matrix. In order to
properly normalize all couplings, we shall express Y as a function of the absolute neutrino
D
mass scale m ∼ 10−2 eV and M ∼ 1014GeV. For this purpose, consider arbitrary M
ν B−L eff
and M and let m and mR be the heaviest mass eigenvalues of M and M , respectively.
R 3 3 eff R
We normalize M by redefining
eff
m
M → M(cid:48) = νM . (12)
eff eff m eff
3
Adjustedinthisway, theheaviestmasseigenvalueofM becomesequaltom . Similarly, we
eff ν
re-scale mR = M to normalize M . The normalization of M in (12) is then equivalent
3 B−L R eff
to the redefinitions
(cid:115)
M m M
M → M(cid:48) = B−LM and M → M(cid:48) = ν B−LM = (cid:104)H0(cid:105)Y , (13)
R R mR R D D m mR D 2 D
3 3 3
where Y is now properly normalized. From (8) and (7a) we find that in this basis
D
M = K∗U(cid:98)†D˜U(cid:98)∗ MdiagK˜U(cid:98)TD U∗. (14)
D R R D(cid:48) D D D (cid:96)
For simplicity, we will make this rescaling in the following without noting the prime in M
R
and M . From (14) and (13), we see that
D
(cid:115)
1 m M
Y = ν B−LK∗U(cid:98)†D˜U(cid:98)∗ MdiagK˜U(cid:98)TD U∗, (15)
D (cid:104)H0(cid:105) m3 mR R R D(cid:48) D D D (cid:96)
2 ν 3
where we have introduced K˜ = K K∗ and D˜ = D∗ D∗. The neutrino Yukawa coupling
D D(cid:48) D(cid:48) R
matrix in (15) can also be written as follows [29]3:
1 (cid:113) (cid:113)
Y = Mdiag ·R· Mdiag ·UT . (16)
D vsinβ R eff PMNS
3Note that the definition of U in [29] differs from our definition by complex conjugation (cf. (11)).
PMNS
5
Here, R denotes a complex orthogonal matrix which may be parametrized in terms of 3
complex angles θ = x +iy as
i i i
 
c c −c s −s s c s s −c s c
2 3 1 3 1 2 3 1 3 1 2 3
R =  c2s3 c1c3 −s1s2s3 −s1c3 −c1s2s3 , (17)
s s c c c
2 1 2 1 2
with (c ,s ) = (cosθ ,sinθ ) = (cosx coshy − isinx sinhy ,sinx coshy + icosx sinhy ).
i i i i i i i i i i i i
The parameters can take the values x ∈ [0,2π[ and y ∈ ]−∞,∞[ (in practical cases,
i i
however, the y are constrained by perturbativity to values |y | (cid:46) O(1), see Sec. 4.2). While
i i
the light neutrino masses m and the mixing angles θ have been measured or constrained,
i ij
the phases φ and δ, the heavy neutrino masses mR and the matrix R are presently unknown.
i i
From (15) and (16) we obtain
R = (Mdiag)−1/2K∗U(cid:98)†D˜U(cid:98)∗ MdiagK˜U(cid:98)TD U(cid:98)∗K∗ (Mdiag)−1/2, (18)
R R R D(cid:48) D D D ν Maj eff
where we have used (10). The parametrization in (16) has the advantage that (i) one can
understand the impact of nonzero CP-violating phases on the LFV rates in a compara-
tively simple way and (ii) one can quickly scan the parameter space while ensuring a valid
low-energy phenomenology (lepton masses and PMNS angles). We will make use of these
properties in subsequent sections. Note, however, that the drawback of the parametrization
(16) is that the exact connection with the lepton mass terms in the Lagrangian has been
lost after rotating to the basis where M and M are diagonal.
(cid:96) R
3 Charged Lepton Flavor Violation in SUSY
Theheavyneutrinomasseigenstatesνc introducedintheseesawmechanismaretooheavyto
i
be observed directly but they influence the mixing of the sleptons in the MSSM via radiative
corrections. The 6×6 slepton mass matrix may be written as a sum of two parts,
(cid:18) m2 m2† (cid:19) (cid:18) δm2 δm2† (cid:19)
m2 = L LR + L LR , (19)
˜l m2 m2 δm2 0
LR R MSSM LR νc
where the first part denoted by MSSM is the usual mass matrix in the MSSM without right-
handed neutrinos. In our analysis of SUSY LFV processes we adopt the mSUGRA scheme
of SUSY-breaking, in which case the slepton mass matrix does not contain flavor mixing
terms. In leading logarithmic approximation the corrections to the slepton mass matrix due
to right-handed neutrinos, denoted by νc in (19), can then be written as [34],
1
δm2 = − (3m2 +A2)Y†LY ,
L 8π2 0 0 D D
3
δm2 = − A vcosβY Y†LY , (20)
LR 16π2 0 l D D
where L = ln(M /mR)δ , mR being the heavy neutrino masses, and m and A are the
ij X i ij i 0 0
universal scalar mass and trilinear coupling, respectively, at M . With the neutrino Yukawa
X
6
Br(µ → eγ) Br(τ → µγ) Br(τ → eγ)
Present 1.2×10−11[30] 6.8×10−8 [31] 1.1×10−7 [32]
Expected 10−13[33] ≈ 10−8 ≈ 10−8
Table 1: Current bounds and expected future sensitivities of direct experimental LFV
searches.
matrix Y and the heavy neutrino masses mR in a given scenario as input at the GUT scale
D i
we calculate the slepton mass matrix at the electroweak scale using (20). The flavor off-
diagonal virtual effects in (20) induced by the mixing in the neutrino sector lead to charged
LFV. More details on this mechanism can be found in [35] and the references therein.
3.1 LFV Rare Decays
In the SUSY seesaw model considered here, LFV processes mainly occur via intermediate
left-handed slepton flavor transitions. The most important low-energy processes are the
rare decays l → l γ,i (cid:54)= j ∈ e,µ,τ, which provide the most stringent bounds on LFV in
i j
the SUSY seesaw model as of now. The current bounds on these processes as well as the
expected sensitivities of future experiments are listed in Table 1.
Each LFV transition is suppressed in a given process by a small factor |(δm2) /m2|2 (i (cid:54)=
L ij (cid:101)
j), where (δm2) are the off-diagonal elements of the left-handed slepton mass matrix m2
L ij L
specifiedin(20)andm2 isoftheorderoftherelevantsparticlemassesintheloopsinvolvedin
(cid:101)
the process. To lowest order in the off-diagonal mass corrections one has approximately [34],
|(δm2) |2
Br(l → l γ) ∝ α3m5 L ij tan2β. (21)
i j li m8
(cid:101)
This expression is just used for illustration. In our numerical calculations we use the full one
loop result for Br(l → l γ), as given in [35].
i j
3.2 LFV Processes at the LHC
At the LHC, a feasible test of LFV is provided by the production of squarks and gluinos,
followed by cascade decays of squarks and gluinos via neutralinos and sleptons [36, 37]:
pp → q˜ q˜,g˜q˜ ,g˜g˜,
a b a
q˜ (g˜) → χ˜0q (g),
a 2 a
χ˜0 → ˜l l ,
2 c i
˜l → χ˜0l , (22)
c 1 j
where a,b,c run over all relevant sparticle mass eigenstates. Lepton flavor violation can
occur in the decay of the second lightest neutralino or the slepton, resulting in different
lepton flavors, i (cid:54)= j. The total cross section for the signature l±l∓+X can then be written
i j
7
as
σ (pp → χ˜0 +X → l±l∓χ˜0 +X)
2 i j 1
(cid:40)
= 2σ(pp → g˜g˜)Br(g˜ → qqχ˜0)
2
(cid:88) (cid:2) (cid:3)
+ σ(pp → g˜q˜ ) Br(g˜ → qqχ˜0)+Br(q˜ → qχ˜0)
a 2 a 2
a
(cid:41)
(cid:88) (cid:2) (cid:3)
+ σ(pp → q˜ q˜) Br(q˜ → qχ˜0)+Br(q˜ → qχ˜0) ×Br(χ˜0 → l±l∓χ˜0), (23)
a b a 2 b 2 2 i j 1
a,b
with Br(g˜ → qqχ˜0) = (cid:80) Br(g˜ → qq˜ )Br(q˜ → qχ˜0) and X can involve jets, leptons and
2 a a a 2
lightest neutralinos produced by lepton flavor conserving decays of squarks and gluinos,
as well as low-energy proton remnants. The LFV branching ratio Br(χ˜0 → l+l−χ˜0) is for
2 i j 1
example calculated in [38] in the framework of model-independent MSSM slepton mixing.
In general, it involves a coherent summation over all intermediate slepton states. In our
numerical calculation we use the leading order partonic cross sections σ(pp → q˜q˜), σ(pp →
q˜g˜) and σ(pp → g˜g˜) folded with the CTEQ6M parton distribution functions together with
(23)inordertocalculatethenumberofeventsfortheLFVprocesspp → χ˜0+X → eµ+χ˜0+X
2 1
expected at the LHC [39, 40].
4 Textures
In the following, we will consider the lepton mass matrix textures from the list of 72 types
givenin[22], wherewenormalizethetexturesaccordingto(13)toobtainleptonandneutrino
masses compatible with experimental data. This list will be called our “reference list” of
textures. Each of the 72 types of textures in the list is characterized by a set {M ,M ,M }
(cid:96) D R
of relevant mass matrices. We call this set of three textures a “texture set”. It is important
to note that the textures M ,M , and M are in general not diagonal and large leptonic
(cid:96) D R
mixing can emerge from any of the matrices M ,M or M . The structure of the non-
(cid:96) D R
diagonal textures is relevant for the construction of explicit high-scale theories, such as the
Froggatt-Nielsen mechanism or discrete flavor models, generating the hierarchical pattern of
these textures. In fact, a large number of explicit models predicting the texture sets from
flavor symmetries have already been found for the SM [23] and also for SUSY SU(5) GUTs
[24].
4.1 Real Textures and Relation to QLC
As shown in [22], each of the 72 texture sets reproduces tribimaximal neutrino mixing in the
neutrino sector at the 3σ level along with the charged lepton mass ratios4
m : m : m = (cid:15)4 : (cid:15)2 : 1, (24)
e µ τ
4We are interested in a fit compatible with minimal SU(5) but different mass spectra, e.g. realization of
the Georgi-Jarlskog relations [41] are possible and can be implemented just as well.
8
and a normal neutrino mass hierarchy of the form
m : m : m = (cid:15)2 : (cid:15) : 1, (25)
1 2 3
where (cid:15) is of the order of the Cabibbo angle (cid:15) (cid:39) θ (cid:39) 0.2 and m = m ≈ 5 × 10−2eV,
C 3 ν
which reproduces the values of the neutrino mass squared differences in (1). In the reference
list, M has the eigenvalues mD and M the eigenvalues mR (i = 1,2,3). Their ratios are
D i R i
always of the form
mD : mD : mD = (cid:15)k : (cid:15)m : (cid:15)n, mR : mR : mR = (cid:15)p : (cid:15)q : 1, (26)
1 2 3 1 2 3
where k,m,n,p,q are non-negative integers ≤ 2 and 0 < p ≤ q. The heavy Majorana
neutrino masses are, thus, always non-degenerate. The textures in the reference list lead to
solar and atmospheric mixing angles that are in agreement with current data at the 3σ level.
Additionally, the reactor angle is very small and satisfies θ < 1◦ which is an attractive
13
parameter range for flavor models although new global fits point slightly towards a nonzero
θ .
13
The textures give rise to a large solar angle θ ≈ 33◦ in a way similar to QLC. This
12
means that the mixing angles entering the mixing matrices U in (8) are either of the order of
x
∼ (cid:15)n, with a positive integer n, or they are equal to π/4, corresponding to maximal mixing.
The nearly tribimaximal mixing form of U , and in particular the observed value of the
PMNS
solar angle, is then a consequence of combining the mixing matrices U from different lepton
x
sectors, such as the charged lepton or left-handed neutrino sector, into U via (10).
PMNS
Lepton mixing angles with positive integer powers of (cid:15) are motivated by the observed
CKM mixing V (cid:39) (cid:15),V (cid:39) (cid:15)2, and V (cid:39) (cid:15)3, and by µ-τ-symmetry [42] (for more recent
us cb ub
studies on µ-τ-symmetry see, e.g. [43]). Differently from most applications of QLC, however,
we do not require that U be of a CKM-like mixing form (i.e. that U (cid:39) V ) or that U
(cid:96) (cid:96) CKM ν
be of the bi-maximal mixing type. Instead, in our reference list, the observed large solar
and atmospheric mixing angles in U can originate from maximal mixing among any
PMNS
two generations of charged leptons, left-handed, or right-handed neutrinos. For instance, we
have in 10% of the cases trimaximal mixing of the left-handed charged leptons, i.e. all three
mixing angles in U are maximal. Such general forms of textures as given in the reference list
(cid:96)
are useful for the construction of new explicit models that can explain the observed lepton
mass and mixing parameters.
Consider, for example, texture set No. 1 from the reference list [22]. The charged lepton
texture reads
 
−a(cid:15)4 0 0
m
M(cid:96) = √τ  a(cid:15)4 (cid:15)2 −(cid:15)2, (27)
2
a(cid:15) 1 1
while the neutrino mass matrix textures are
   
(cid:15)2 a(cid:15)2 −(cid:15)2 b(cid:15)2 a(cid:15)3 a(cid:15)3
mD mR
MD = −√3 a(cid:15) 1 (cid:15)2 , MR = 3 a(cid:15)3 1+(cid:15) −1+(cid:15), (28)
2 2
a(cid:15)4 −1 a(cid:15) a(cid:15)3 −1+(cid:15) 1+(cid:15)
9
√
where the order-one coefficients a and b, take the values a = 2, and b = 2. The mass
parameter mD ∼ 102GeV is determined by a normalization as described in (13). For this
3
example, the ratios of the corresponding eigenvalues are
mD : mD : mD = (cid:15)2 : 1 : (cid:15), mR : mR : mR = (cid:15)2 : (cid:15) : 1. (29)
1 2 3 1 2 3
The charged lepton and light neutrino mass ratios are as in (24) and (25). This texture set is
an example where we have maximal mixing among the 2nd and 3rd generation of left-handed
charged leptons, θ(cid:96) = π/4, and right-handed neutrinos, θR = π/4 (in the notation of Sec. 2)
23 23
[22]. Note also that in M it is the 2nd column that is dominant. The PMNS mixing angles
D
for this texture set are given by5
π (cid:15) π (cid:15) (cid:15)2
θ = − √ +O((cid:15)2), θ = + √ +O((cid:15)2), θ (cid:39) . (30)
12 23 13
4 2 4 2 2
Notethatthistexturesethastheinterestingpropertythatθ issuppressedbytwopowersof
13
the Cabibbo angle. We have checked numerically that this is a stable feature under variation
of the lepton Yukawa couplings and mass terms. After rotating to the basis where M is
(cid:96)
diagonal, the normalized neutrino Yukawa coupling matrix is to leading order in (cid:15) given by
   
(cid:15)2 −1.7(cid:15)2 −0.28(cid:15)2 (cid:15)2 (cid:15)2 (cid:15)2
(m mR)1
YD = 0.045 (cid:15) 0.71(cid:15) −0.71(cid:15) (cid:39) ν 3 2 (cid:15) (cid:15) (cid:15). (31)
vsinβ
−0.28(cid:15)2 −1 −1 (cid:15)2 1 1
wherewehavetakenm = 5×10−2eV,mR = 2.5×1012GeVandtanβ = 10. Withthischoice
ν 3
the other heavy right-handed neutrino masses are mR = 1011GeV and mR = 5×1011,GeV.
1 2
Thedimensionlesscoefficient0.045multiplyingthematrixin(31)issetbythescalesm ,mR,
ν 3
and vsinβ. Note that even after rotating to this basis, the factors multiplying within the
matrix the powers of (cid:15) are all approximately of order one.
4.2 Complex Textures
The reference list contains only real matrices and therefore describes only CP-conserving
cases. We introduce CP-violation for the different texture sets by multiplying each mass
matrix element in the Lagrangian in (5) by an individual random phase. This means that
we modify for the texture set {M ,M ,M } each mass matrix element to
(cid:96) D R
(M ) → (M(cid:48)) = exp(iαx)·(M ) (x = (cid:96),D,R), (32)
x ij x ij ij x ij
where we assume that the 9+9+6 = 24 phases αx vary independently on a linear scale over
ij
thewholeintervalαx ∈ [0,2π[. Noticethatonecanrotateaway10ofthe24phasesbyphase-
ij
redefinitions, but we will not make use of this freedom here. Moreover, the mapping in (32)
leaves the moduli of the mass matrix elements invariant but changes the matrix R in (17),
the PMNS mixing parameters, and the Dirac and Majorana phases δ,φ , φ . For each of the
1 2
5Notethatfutureexperimentalboundsmightaffordittoconsidercorrectionstothesolarandatmospheric
mixing angle stemming e.g. from the charged lepton sector.
10