Lepton mixing and the chargedlepton mass ratios
HJE
mixing and the chargedlepton mass ratios
Darius Jurˇciukonis 0 1 3
Lu´ıs Lavoura 0 1 2
0 1049001 Lisboa , Portugal
1 Saule ̇tekio ave. 3, LT10222 Vilnius , Lithuania
2 Instituto Superior T ́ecnico, CFTP, Universidade de Lisboa
3 Institute of Theoretical Physics and Astronomy, University of Vilnius
We construct a class of renormalizable models for lepton mixing that generate predictions given in terms of the chargedlepton mass ratios. We show that one of those models leads, when one takes into account the known experimental values, to almost maximal CP breaking phases and to almost maximal neutrinoless doublebeta decay. We study in detail the scalar potential of the models, especially the bounds imposed by unitarity on the values of the quartic couplings.
Phenomenological Models

Lepton
2
3
4
2.1
2.2
4.1
4.2
4.3
4.4
4.5
4.6
4.7
1 Introduction and notation
Models A specific model The scalar potential
Procedure for producing the scatter plots
Scatter plots
5
Conclusions
A Unitarity bounds for a 3HDM with Z2 × Z2 × Z2 symmetry
A.1 General case
A.2 Case with additional symmetry φ2 ↔ φ3
B Other stability points
1
Model (SM). The models have gauge group SU(2) × U(1) just as the SM. They feature an
extended scalar sector, with three SU(2) doublets φk = φk , φ
k
+
0 T (k = 1, 2, 3) instead of
one; we let φ˜k = φ0k∗, −φk
− T denote the conjugate doublets. The leptonic sector is also
extended, with the addition to the SM of three righthanded (i.e. SU(2)singlet) neutrinos
νRψ,
1 which enable a typeI seesaw mechanism [1–5] to suppress the standardneutrino
masses. Our models have familyleptonnumber symmetries
DLψ → eiξψ DLψ,
ψR → eiξψ ψR,
νRψ → eiξψ νRψ,
(1.1)
1In this paper the Greek letters ψ, α, β, and γ in general run over the lepton flavours e, μ, and τ .
Whenever we use (α, β, γ) we mean a permutation of (e, μ, τ ), i.e. α 6= β 6= γ 6= α.
– 1 –
where
given by
are the chargedlepton masses. The symmetries (1.1) leave the Yukawa couplings invariant
but they are broken softly by the Majorana mass terms of the righthanded neutrinos,
Mℓ = diag (ℓe, ℓµ , ℓτ ) ,
MD = diag (De, Dµ , Dτ ) ,
ℓψ = mψ
LMaj = − 2
1
νeR, νµR , ντR
νeRT
MRC νµR T + H.c.
ντRT
where the phases ξe, ξµ , and ξτ are arbitrary and uncorrelated. In transformation (1.1),
ψR denotes the righthanded charged leptons and DLψ = (νLψ, ψL)
doublets of lefthanded neutrinos νLψ and charged leptons ψL. In our models both the
chargedlepton mass matrix Mℓ and the neutrino Dirac mass matrix MD are diagonal,
because they originate in Yukawa couplings that respect the familyleptonnumber
sym
T denotes the SU(2)
metries (1.1). Thus,
In equation (1.4), C is the chargeconjugation matrix in Dirac space; the 3×3 flavourspace
matrix MR is symmetric. The seesaw mechanism produces an effective lightneutrino mass
matrix M = −MDMR−1MD, i.e.
Mψψ′ = −DψDψ′ MR−1
ψψ′ ,
∀ ψ, ψ′ ∈ {e, µ, τ } .
Note that, since MD and Mℓ are diagonal, the matrix MR is the sole origin of lepton mixing
in our models [6, 7].2 The symmetric matrix M is diagonalized as
U T M U = diag (m1, m2, m3) ,
where the mk are the (nonnegative real) lightneutrino masses and U is the lepton mixing
matrix, for which we use the standard parameterization [9]
(1.2)
(1.3)
(1.4)
(1.5)
(1.6)
(1.7)
HJEP03(218)5
c12c13
s12c13
U = −s12c23 − c12s23s13eiδ c12c23 − s12s23s13eiδ
× diag 1, eiα21/2, eiα31/2 ,
where cij = cos θij and sij = sin θij for ij = 12, 23, 13. The phase δ is the Dirac phase; α21
and α31 are the Majorana phases.
2In the study of leptogenesis one uses a basis for the mass matrices where Mℓ and MR are diagonal but
MD is not. In our models Mℓ and MD are diagonal but MR is not. If in our models MR is diagonalized
as V T MRV = diag (M1, M2, M3), where V is a 3 × 3 unitary matrix and M1,2,3 are nonnegative real, then
M D′ = V T MD in the basis appropriate for the computation of leptogenesis. The Hermitian matrix relevant
for leptogenesis is R ≡ M D′M D′† = V T MDM D†V ∗, which is nondiagonal. Thus, leptogenesis is in principle
viable in our models. See ref. [8] for details.
– 2 –
The purpose of our models is to make predictions for the matrix U . There are in the
literature many predictive models for U ;3 the models in this paper are original in that
they are welldefined renormalizable models that produce predictions for the neutrino mass
matrix M in terms of chargedlepton mass ratios. Since the mass ratios mµ /mτ , me/mµ ,
and me/mτ are very small, the predictions of our models are hardly distinguishable in
practice from the cases with ‘texture zeroes’ in the neutrino mass matrix [
13
].
In section 2 we expound the construction of the models and classify the various models
that our class of models encompasses. Section 3 focusses on a specific model with
remarkable predictions: almostmaximal δ and almostmaximal neutrinoless doublebeta decay.
Section 4 discusses a scalar potential for our models and the way in which that potential is
able to reproduce the Higgs particle discovered at the LHC. Section 5 contains our main
conclusions. Appendix A deals on the derivation of the unitarity bounds on the coupling
constants of the scalar potential. In appendix B we compute the expectation value of the
scalar potential in the various stability points of that potential.
2
2.1
Models
Construction of the models
1. Models with Yukawa Lagrangian
Our class of models may be divided into four subclasses:
where (α, β, γ) is a permutation of (e, µ, τ ) and y1,2,3,4 are Yukawa coupling
constants, which are in general complex.
2. Models with Yukawa Lagrangian
L1 = −y1DLανRαφ˜1 − y2DLααRφ1
− y3 DLβνRβφ˜2 + DLγ νRγ φ˜3
− y4 DLββRφ2 + DLγ γRφ3 + H.c.,
L2 = −y1DLανRαφ˜1 − y2DLααRφ1
− y3 DLβνRβφ˜2 + DLγ νRγ φ˜3
− y4 DLββRφ3 + DLγ γRφ2 + H.c.
Note that L1 and L2 differ only in their last lines.
3. Models with Yukawa Lagrangian
L3 = −y1DLανRαφ˜1 − y2DLααRφ1
− y3DLβνRβφ˜2 − y3∗DLγ νRγ φ˜3
− y4DLββRφ2 − y4∗DLγ γRφ3 + H.c.,
where y1 and y2 are real while y3 and y4 are in general complex.
3See the reviews [10–12]; the original papers are in the bibliographies of those reviews.
– 3 –
(2.1)
(2.2)
(2.3)
4. Models with Yukawa Lagrangian
L4 = −y1DLανRαφ˜1 − y2DLααRφ1
− y3DLβνRβφ˜2 − y3∗DLγνRγφ˜3
− y4DLββRφ3 − y4∗DLγγRφ2 + H.c.,
where once again y1 and y2 are real. The Lagrangians (2.3) and (2.4) differ in their
last lines.
It is clear that L1,2,3,4 enjoy the familyleptonnumber symmetries (1.1). The
Lagrangians (2.1) and (2.2) further enjoy the interchange symmetry
The Lagrangians (2.3) and (2.4) are invariant under the CP symmetry
φ1 (x) → φ1∗ (x¯) ,
αR (x) → K αRT (x¯) ,
φ2 (x) → φ3∗ (x¯) ,
βR (x) → K γRT (x¯) ,
φ3 (x) → φ2∗ (x¯) ,
γR (x) → K βRT (x¯) ,
νRα (x) → K νRαT (x¯) , νRβ (x) → K νRγT (x¯) , νRγ (x) → K νRβT (x¯) ,
DLα (x) → KDLαT (x¯) , DLβ (x) → KDLγT (x¯) , DLγ (x) → KDLβT (x¯) ,
where x ≡ (t, ~r) and x¯ ≡ (t, −~r); K ≡ iγ0C is the CP transformation matrix in Dirac
space. Moreover, in the last line of transformation (2.6),
T
DLψ ≡
νLψT !
ψLT
.
The CP transformation (2.6) interchanges the lepton flavours β and γ.
The Lagrangians (2.1)–(2.4) necessitate additional symmetries to guarantee that each
scalar doublet only couples to the desired lepton flavour. There is a large arbitrariness in
the choice of the additional symmetries. In this paper we choose them to be
for all four Lagrangians (2.1)–(2.4); and either
for Lagrangians (2.1) and (2.3), or else
Z(22) : φ2 → −φ2, βR → −βR, νRβ → −νRβ,
Z(23) : φ3 → −φ3, γR → −γR, νRγ → −νRγ,
Z(24) : φ2 → −φ2, γR → −γR, νRβ → −νRβ,
Z(25) : φ3 → −φ3, βR → −βR, νRγ → −νRγ,
for Lagrangians (2.2) and (2.4). The transformations (2.8) and either (2.9) or (2.10) form
a Z2 × Z2 × Z2 symmetry.
– 4 –
(2.4)
(2.5)
(2.6)
(2.7)
(2.8)
(2.9)
(2.10)
Let vk denote the vacuum expectation value (VEV) of φ0k. Then, from L1,
(Mℓ)αα ≡ ℓα = y2v1,
(Mℓ)ββ ≡ ℓβ = y4v2,
(Mℓ)γγ ≡ ℓγ = y4v3,
(MD)αα ≡ Dα = y1∗v1, (MD)ββ ≡ Dβ = y3∗v2, (MD)γγ ≡ Dγ = y3∗v3
for model 1. From the Yukawa Lagrangian (2.2),
for model 2. From L3,
for model 3. From the Yukawa Lagrangian (2.4),
assume that they do not break either the interchange symmetry (2.5) of models 1 and 2
or the CP symmetry (2.6) of models 3 and 4. This means that, in models 1 and 2,
(MR)ββ = (MR)γγ , (MR)αβ = (MR)αγ .
Clearly, the symmetry (2.15) for the matrix MR is also valid for the matrix MR−1. Therefore,
from equation (1.5),
for models 1 and 2. This means that the rephasinginvariant phase
Mββ =
Mγγ
Dβ
Dγ
2
,
Mαβ =
Mαγ
Dβ
Dγ
arg hMγγ (Mαβ)2 Mβ∗β Mα∗γ 2i = 0
in models 1 and 2. Additionally, from equations (2.11) and (1.3),
for model 1; while, from equations (2.12) and (1.3),
for model 2.
Mββ
Mγγ
Mββ
Mγγ
=
=
m2β
m2γ
m2γ
m2
β
,
,
Mαβ
Mαγ
Mαβ
Mαγ
=
=
mβ
mγ
mγ
mβ
– 5 –
(2.11)
(2.12)
(2.13)
(2.14)
(2.15)
(2.16)
(2.17)
(2.18)
(2.19)
HJEP03(218)5
We conclude that model 1 makes three predictions for the effective lightneutrino mass
matrix M : equations (2.17) and (2.18).
Model 2 also makes three predictions:
equations (2.17) and (2.19).
In models 3 and 4, we assume that the CP symmetry (2.6) is not broken by the
Majorana mass terms of the νR. This means that
(MR)ββ = (MR∗ )γγ , (MR)αβ = (MR∗ )αγ ,
(MR)αα = (MR∗ )αα , (MR)βγ = (MR∗ )βγ
(2.20)
(2.21)
(2.22a)
(2.22b)
in those models. Equations (2.20) are valid for MR−1 as weall as for MR, hence
,
Mαβ =
Mα∗γ
DαDβ
Dα∗Dγ∗
Mαα =
Mα∗α
DαDα
Dα∗Dα∗
,
Mβγ =
Mβ∗γ
DβDγ
Dβ∗ Dγ∗
for models 3 and 4. Equations (2.21) imply the following rephasinginvariant conditions on
the matrix M :
arg hMβ∗βMγ∗γ (Mβγ )2i = 0,
arg Mα∗αMβ∗γ MαβMαγ = 0.
Moreover, from equations (1.3) and (2.13) one derives equation (2.18), which is thus also
valid for model 3; from equations (1.3) and (2.14) one derives equation (2.19), which
thus applies to model 4. We conclude that model 3 makes four predictions for M :
equations (2.22) and (2.18). Model 4 also makes four predictions: equations (2.22) and (2.19).
2.2
Classification of the models
Our class of models encompasses twelve models, depending on whether one uses model 1,
2, 3, or 4 and depending on whether the flavour α is taken to be e, µ , or τ . (The flavours
β and γ are treated symetrically in the models.)
There is a distinction between the models with interchange symmetry (2.5) and the
models with CP symmetry (2.6): the former lead to only one constraint (2.17) on the
phases of the matrix elements of M , while the latter lead to the two constraints (2.22).
The CP symmetry (2.6) is more powerful than the interchange symmetry (2.5).
However, in practice the distinction between equation (2.17) and equations (2.22) is not
very significant, because the chargedlepton mass ratios are so small that they force some
M matrix elements to be very close to zero, hence their phases do not matter much. We
see from equations (2.18) and (2.19) that our twelve models may be classified in six types:
i. Models that predict
Mee
Mµ
=
me2
mµ2
,
Meτ
Mµτ
=
me
mµ
Since me ≪ mµ , in these models one is close to the situation Mee = Meτ = 0, which
is case A2 of ref. [
13
].
– 6 –
ii. Models that predict
iii. Models that predict
to case A1 of ref. [
13
].
iv. Models that predict
v. Models that predict
vi. Models that predict
Mee
Mµ
Mee
Mττ
Mee
Mττ
Mµ
Mττ
Mµ
Mττ
=
=
=
=
=
mµ2
me2
me2
mτ2
mτ2
me2
mµ2
mτ2
mτ2
mµ2
,
,
,
,
,
Meτ
Mµτ
=
mµ
me
.
Meµ
Mµτ
=
me
mτ
.
Meµ
Mµτ
Meµ
Meτ
Meµ
Meτ
=
=
=
mτ
me
mµ
mτ
mτ
mµ
.
≈ 0 and Meµ ≈ 0. They are therefore
close to case B3 of ref. [
13
].
Since me ≪ mµ these models predict Mµ ≈ 0 and Mµτ ≈ 0. According to ref. [13],
Since me ≪ mτ these models predict Mee ≈ 0 and Meµ ≈ 0. They are therefore close
This leads to Mττ ≈ 0 and Meτ ≈ 0, corresponding to case B4 of ref. [
13
].
We thus find that, out of our twelve models, four should be phenomenologically
excluded. The other eight are viable; two of them approximately coincide in their predictions
with case A1 of ref. [
13
], two other with case A2, two more with case B3, and the last two
with case B4.
We have made numerical simulations of all our models and they very much vindicate
the above conclusions. We do not feel it worth presenting those numerical simulations in
detail here. In the next section we focus solely on one model that in our opinion yields
particularly interesting results.
3
A specific model
In this section we deal on one of our models, which predicts
Mµ
Mττ
Meµ
Meτ
=
=
mµ2
mτ2
mµ
mτ
arg hMττ (Meµ )2 Mµ ∗ (Me∗τ )2i = 0.
– 7 –
(2.24)
(2.25)
(2.26)
(2.27)
(2.28)
(3.1a)
(3.1b)
(3.1c)
Equations (3.1) are three predictions. This is not much; for instance, each of the cases with
two texture zeroes of ref. [
13
] has four predictions, and there are models with as many as
six predictions for M . So, one might think that the predictions (3.1) are of little practical
consequence. This is not so, however.
We use M = U ∗ diag (m1, m2, m3) U † and the parameterization of U in equation (1.7).
We also use the experimental 3σ bounds [
14
]
and either
for a normal ordering of the neutrino masses, or
95% confidence level [15].4
for the inverted ordering of the neutrino masses. The phases δ, α21, and α31 are
unknown, just as the overall scale of the neutrino masses; we represent the latter through
msum ≡ m1 + m2 + m3. Strong cosmological arguments suggest that msum ≤ 0.25 eV at
A quantity of especial importance is
mββ ≡ Mee = m1c122c123 + m2s122c123eiα21 + m3s123ei(α31−2δ) .
This quantity is relevant for neutrinoless doublebeta decay, which should proceed with a
rate approximately proportional to m2ββ. It is clear that mββ becomes maximal when
for whatever value of the phase δ.
In figure 1 we have plotted mββ as a function of msum, both when only the
inequalities (3.2) and either (3.3) or (3.4) hold, and when furthermore the predictions (3.1) are
enforced. The information in that figure is clear: the predictions (3.1) lead to almost
maximal mββ, irrespective of the neutrino mass ordering.
This of course happens because equations (3.6) hold. In figure 2 one observes that this
is indeed so and that, moreover, the predictions (3.1) lead to δ ≈ 3π/2. Thus, our model
firmly predicts the three phases δ, α21, and α31; the phase δ is predicted to be very close
to 1.5π, and this agrees nicely with its 1σpreferred experimental value [
14
].
4A recent paper [
16
] claims that msum = 0.11 ± 0.03 eV.
– 8 –
(3.2a)
(3.2b)
(3.2c)
(3.3a)
(3.3b)
(3.4a)
(3.4b)
(3.5)
(3.6)
trino masses and figure 1b is for an inverted ordering. The red points take into account only the
experimental bounds (3.2)–(3.4); the blue points arise from the constraints (3.1).
is for a normal ordering of the neutrino masses and figure 2b is for an inverted ordering.
One moreover observes in figure 1 that our model does not tolerate very low neutrino
masses, but goes well with almostdegenerate neutrinos: msum & 0.15 eV for both the
normal and inverted neutrino mass spectra.
This specific model does not just predict the Dirac and Majorana phases; it moreover
predicts the quadrant of the angle θ23 and a correlation between that angle and msum.
That is observed in figure 3. One sees that θ23 lies in the first quadrant when the neutrino
mass ordering is normal, in the second quadrant when it is inverted. One also sees that
θ23 is correlated with msum, with θ23 becoming ever closer to π/4 when msum grows.
Figures 1 and 3 are very similar to analogous figures displayed in ref. [17] for case B3
of ref. [
13
]. That case is defined by Mµ
= Meµ = 0,5 which of course means four
pre5The paper of ref. [
13
] contains various twotexturezero cases, in particular case B3 defined as
Mµ
= Meµ = 0.
The cases are of course not full models.
However, it was demonstrated in ref. [
18
]
that any texturezero mass matrix may result from a renormalizable model.
normal ordering of the neutrino masses and figure 3b is for an inverted ordering.
dictions for M (because both the moduli and phases of Mµ
and Meµ are relevant). Our
predictions (3.1) mean that our model features both Mµ  ≪ Mττ  and Meµ  ≪ Meτ ,
and this is an approximation to case B3. As a matter of fact, we have explicitly checked
that the two conditions (3.1a) and (3.1b) by themselves alone lead to almost the same
allowed domains as in figures 1–3, and as in case B3 of ref. [
13
]. The two conditions (3.1a)
and (3.1b) are in practice just as predictive as that case with four predictions.
4
4.1
The scalar potential
Assumptions
In this section we investigate a way in which our class of models with three Higgs doublets
and various symmetries may (i) be extended to the quark sector, and (ii) produce scalar
particles with masses and couplings in agreement with the phenomenology. The aim of
our investigation is to demonstrate that this can be done; we do not explore the full set of
options. Thus, in this section we make additional assumptions. We stress that the validity
of the models expounded in section 2 is in general independent of the specific additional
assumptions that we shall utilize in this section.
Our main assumption is that there are no scalars besides the three Higgs doublets that
have Yukawa couplings to the leptons. Therefore,
v ≡
qv12 + v22 + v32 =
√
2mW
g
≈ 174 GeV,
(4.1)
where mW = 80.1 GeV is the mass of the W ± bosons and g is the gaugeSU(2) coupling
constant.
Our models have either an interchange symmetry (2.5) or the CP symmetry (2.6).
Those symmetries are unbroken by the Majorana mass terms of the righthanded neutrinos,
which have mass dimension three. Still, those symmetries may be broken by the quadratic
(i.e. mass dimension two) terms of the scalar potential. We shall assume that this does not
happen, i.e. that either the interchange symmetry (2.5) or the CP symmetry (2.6) are
conserved by the quadratic terms of the scalar potential. The potential is thus symmetric under
either φ2 ↔ φ3 or φ2 ↔ φ3∗. In this paper we shall only consider the potential invariant
under φ2 ↔ φ3;6 in ref. [
19
] the potential invariant under φ2 ↔ φ∗3 has been studied.
Besides, the models have additional symmetries (2.8) together with either (2.9)
or (2.10).7
The symmetry (2.8) does not involve the νRψ and is therefore unbroken by
LMaj. We shall assume that it is also unbroken by the scalar potential; thus, the
potential is invariant under Z(1) : φ1 →
2
−φ1.8
Z(2,4) : φ2 → −φ2 and Z(3,5) : φ3 → −φ3 in the scalar sector, are softly broken by LMaj,
2 2
which is of dimension three; therefore, they must also be broken in the quadratic part of
The symmetries (2.9) or (2.10), which read the potential. The potential therefore is
The parameters µ 3 and λ8 are real because of the symmetry under φ2 ↔ φ3. We use the
freedom of rephasing φ1 to set λ7 real too.
4.2
The vacuum
We assume as usual that the vacuum does not break electromagnetic invariance, i.e. that
the upper components of φ1,2,3 have zero VEV.
The potential (4.2) may, at least for some values of its parameters, produce stability
points with nontrivial relative phases among the VEVs. Those stability points are,
unfortunately, hard to manipulate analytically. We shall neglect them and assume that the
6Our potential is therefore invariant under a Z2 symmetry. When that Z2 symmetry is spontaneously
broken, the vacuum is twofold degenerate. There is a minimumenergy field configuration that interpolates
between the two different vacua; this is called a domain wall. The nonobservation of domain walls definitely
is a problem for our potential. However, we recall the reader that our analysis only purports to display
a particularly simple and illustrative case; we claim our potential neither to be realistic nor to be unique.
The validity of the models expounded in section 2 is independent of the specific scalar potential that we
analyze in this section.
7The additional symmetries are largely arbitrary — in the construction of the models we might have
chosen different additional symmetries to the same practical effect, viz. preventing each scalar doublet from
having Yukawa couplings to more than one lepton doublet. Each specific additional symmmetry alters
the scalar potential in a different way. Thus, in a sense the specific additional symmetries (2.8)–(2.10)
constitute an assumption of this section.
8In ref. [
19
] a potential with quadratic terms φ†1φ2, φ†1φ3, and their Hermitian conjugates has been
analyzed. The fit in this section does not allow for those terms, which break the symmetry Z(21). See also
footnote 6.
three VEVs vk ≡ 0 φ0k 0 are (relatively) real. The VEV of the potential is then
The equations for vacuum stability are
We want a vacuum state with v1 6= 0, because in our models one of the chargedlepton
masses is proportional to v1. We also want the vacuum to have v2 6= v3, because in our
models r ≡ v2/v3 is equal to a ratio of chargedlepton masses. Fortunately, equations (4.5)
have a solution with v1 6= 0 and v2 6= ±v3:
µ 1 = −2λ1v12 − 2l3 v22 + v32 ,
µ 2 = −2l3v1 − 2λ2 v22 + v32 ,
2
µ 3 = 2 (λ2 − l4) v2v3.
Plugging equations (4.6) into equation (4.3), we obtain
V0 =
,
and we use r = mµ /mτ (the results for either r = me/mµ or r = me/mτ are not
qualitatively different). The angle β will be taken to lie in the first quadrant. In this way v1 and
v3 are positive, but this represents no lack of generality. Only the relative sign of v2 and v3
matters, and we have found out that the best results are obtained when v2v3 is negative.
(4.3)
(4.4a)
(4.4b)
(4.5a)
(4.5b)
(4.5c)
(4.6a)
(4.6b)
(4.6c)
(4.7)
(4.8a)
(4.8b)
(4.8c)
We expand the neutral components of the doublets as
where the fields ρk and ηk are real. Subsuming the terms of the potential quadratic in the
Mφ φ2++ ,
φ
3
0 + (2λ2 − λ4 − λ6 + 2λ8) 0 −v32 v2v3 ,
(4.10a)
we find, by using equations (4.6), that
In general, the matrices Mη and Mφ must have an eigenvector (v1, v2, v3) with
eigenvalue zero, corresponding to the Goldstone bosons, hence they must be of form
Mη,φ = aη,φ −v1v2
0 + bη,φ
+ cη,φ 0
v
2
2
0
In our specific case, due to the φ2 ↔ φ3 symmetry of V , the coefficients aη,φ = bη,φ. This
has the important consequence that both Mη and Mφ are diagonalized by the orthogonal
matrix
where
where v23 ≡ pv22 + v32 = v cos β. We find that
We diagonalize Mρ as
v1/v
0
−v23/v
Ov = v2/v v3/v23 v1v2/ (vv23) ,
where Oρ is a real, orthogonal matrix. We order its columns in such a way that m2H1 ≤
m2H2 ≤ m2H3. The fields Hk = P3k′=1 ρk′ (Oρ)k′k are physical scalars with mass mHk .
4.4
The oblique parameter T
the oblique parameter T is [20]
F (x, y) ≡
x + y
0
2
xy
− x − y
ln
x
y ⇐ x 6= y,
⇐ x = y,
T =
1
3
16πs2wm2W
3
+ X
3
X
k=2 k′=1
+ 3 X
k=1
where mZ is the Zboson mass, mW is the W boson mass, mH is the reference mass of the
Higgs boson (which is taken to be 125 GeV), and s2w = 1 − m2W /m2Z . According to ref. [9],
−0.04 < T < 0.20.
4.5
There are many possible ways of extending our models to the quark sector. If one envisages
a model with the CP symmetry (2.6), then that symmetry must be broken spontaneously
through v2 6= v3∗ and that breaking must be felt in the quark sector, because we know that
there is CP violation in that sector; this can be achieved only if both scalar doublets φ2 and
φ3 have Yukawa couplings to the quarks. In a model with the interchange symmetry (2.5),
on the other hand, CP violation may proceed through complex Yukawa couplings and it
is not necessary for φ2 and φ3 to couple to the quarks. Things then become much simpler
because at tree level there are no flavourchanging neutral currents mediated by the neutral
scalars and therefore the neutral scalars do not need to be so heavy. Thus, we extend the
symmetry Z(21) of equation (2.8) as
where the QLk are the gaugeSU(2) doublets of lefthanded quarks. With this extended
Z(21), the quarks only couple to φ1. The Yukawa couplings of the quarks are then given by
Lquark Yukawa =
χ mχ
√2v1
−ρ1 + iη1γ5 χ −
X
ζ=d,s,b
ζ mζ
ρ1 + iη1γ5
√2v1
X
χ=u,c,t
ϕ+
1 X X
+ v1 χ=u,c,t ζ=d,s,b
= − X
k=1
3 Hk (Oρ)1k
√2v sin β
+
G0 − A3 cot β
√2v
+
G+ −ϕ3+ cot β
v
X
χ=u,c,t
X
χ=u,c,t
ζ
Vχζ χ (mχPL − mζ PR) ζ + H.c.
mχ χχ +
mχ χiγ5χ −
X
ζ=d,s,b
X
ζ=d,s,b
mζ ζζ
mζ ζiγ5ζ
(4.20a)
(4.20b)
X
X
χ=u,c,t ζ=d,s,b
Vχζ χ (mχPL −mζ PR) ζ + H.c., (4.20c)
where PR,L are the projectors of chirality, G0 is the neutral Goldstone boson, G± are the
charged Goldstone bosons, A3 is a physical pseudoscalar with mass mA3, and ϕ
physical charged scalars with mass mϕ3. Notice in lines (4.20a) and (4.20b) the absence of
3± are the
flavourchanging couplings of the neutral scalars.
4.6
Procedure for producing the scatter plots
The input for our scatter plots is β and the eight λp (p = 1, . . . , 8).
In inequalities (4.21),
where Θ is the step (Heaviside) function.
the λp, which are derived in appendix A:
In order for the potential not to break unitarity we impose the following conditions on
In order for the potential to be bounded from below we require that the λp satisfy [21]
The angle β is an input of our scatter plots. The VEVs v1,2,3 are determined from
equations (4.8), where v is given by equation (4.1) and r = mµ /mτ . Then, µ 1, µ 2, and
µ 3 are computed by using equations (4.6). The value of V0 is given by equation (4.7). We
require V0 < 0. We also enforce a number of conditions related to the alternative stability
points in appendix B:
require V0 < V0(1±), where the quantities V0(1±) are given in equation (B.2).
• If the quantities in the righthand sides of equations (B.1) are both positive, then we
V (5±) are given in equation (B.11).
V0 < V0(3), where V0(3) is given in equation (B.5).
• If the quantity in the righthand side of equation (B.4) is positive, then we require
V0 < V0(4), where V0(4) is given in equation (B.9).
• If the quantity in the righthand side of equation (B.8a) is positive, then we require
• If the quantity in the righthand side of equation (B.10) is positive (with either the
plus or the minus sign), then we require V0 < V (5±) (with the same sign), where
0
• If the quantity in the righthand side of equation (B.12) is positive and the
inequality (B.14) is satisfied, then we require V0 < V0(6), with V0(6) given in equation (B.13).
We compute the squared masses in equations (4.15).
We construct Mρ in equa
tion (4.11c) and diagonalize it according to equation (4.16).
We assume that the lightest physical scalar, viz. H1, corresponds to the scalar particle
discovered at LHC ; we therefore fit its mass mH1 to be 125 GeV. This fit is very precise,
hence mH1 never needs to appear in our scatter plots.
We require that the masses of the six additional scalars, i.e. mϕ2,3 , mA2,3 , and mH2,3 ,
are all larger than 150 GeV. We also require the parameter T , computed through
equation (4.18), to lie in between −0.04 and +0.20 [9].
The particle discovered at LHC, which we interpret as our H1, couples to gaugeboson
pairs, to the heavy quarks, and to the τ lepton with strengths close to the predictions of
the SM. We hence derive the following constraints:
• The strength of the coupling of H1 to gaugeboson pairs, divided by the strength of
the coupling of the SM Higgs boson to gaugeboson pairs, is [20] gZZ ,9 where
In our scatter plots we demand that 0.9 < gquarks < 1.1.
9The important quantity is gZZ, not gZZ itself, because the sign of the first column of the matrix Oρ is
arbitrary and physically meaningless, hence the sign of gZZ is also arbitrary. Alternatively, we may reason
that the physical cross sections depend on the squared amplitudes, hence on gZ2Z, not on the amplitudes
themselves.
gZZ ≡ v
1
3
k=1
X vk (Oρ)k1 .
gZZ  > 0.9.
gquarks ≡ sin β gZZ 
(Oρ)11 gZZ .
Note that −1 ≤ gZZ ≤ 1, because gZZ is the scalar product of two unit vectors. The
limit gZZ  = 1 corresponds to H1 coupling to pairs of gauge bosons with exactly the
same strength as the SM Higgs boson does. In our scatter plots we require
• We observe in equation (4.20a) that H1 couples to the quarks with strength
(Oρ)11 sin β times the strength of the coupling to the quarks of the SM Higgs boson.
Since the sign of (Oρ)11 is physically meaningless but is correlated with the sign of
gZZ , we define
(4.24)
(4.25)
(4.26)
• We use v2/v3 = mµ /mτ ; this means that we are assuming that, in our specific
model, it is the scalar doublet φ3 that couples to DLτ τR. Thus, there is a Yukawa
coupling
The modulus of the Yukawa coupling constant Υ of course is mτ /v3. Since
H1 couples to τLτR with strength (O√ρ)231v Υ.Therefore, for H1 to couple to τ leptons
2 . The modulus of the coupling of
the SM Higgs boson to τLτR is mτ
with the same strength as the SM Higgs boson, one needs to have (Oρ)31 v3 ≈ 1/v.
Defining
we demand that 0.9 < gτ < 1.1.
gτ ≡
(Oρ)31 v gZZ ,
v3
gZZ 
Furthermore, we see in equation (4.20c) that the physical charged scalars ϕ3± interact
with the quarks in the same way as the charged scalars of the typeI twoHiggsdoublet
model. Therefore, in our scatter plots we have borrowed the bounds in the tan β–mϕ3
plane given in figure 18 of ref. [22].
4.7
Scatter plots
In figure 4 we plot the mass of the lightest new scalar, i.e. of H2, against β. One sees that β
must always be close to 45◦ and that β becomes ever more restricted when the newscalar
masses get higher. Also notice that mH2 cannot be much higher than 300 GeV.
In figure 5 we plot tan β against the mass of the physical charged scalars ϕ3± that
interact with the quarks. Also marked in figure 5, through a solid line, is the phenomenological
scalars ϕ
lower bound on the mass of ϕ3±, which we have taken from figure 18 of ref. [22]. That bound
incorporates the constraints from Z → b¯b, ǫK , and ΔmBs; it guarantees that the charged
3± do not mediate excessively strong ΔS = 2 transitions through box diagrams.
In figure 6 we plot the quantities defined in equations (4.26) and (4.29) against each
other. They seem to be anticorrelated; the anticorrelation becomes more welldefined
when the masses of all the new scalar particles are higher.
In figure 7 we plot the eight parameters λp of the scalar potential. One observes that
λp is never larger than 2 for p ∈ {1, 2, 7, 8}; for 3 ≤ p ≤ 6 the λp may be somewhat larger.
In figure 8 we have plotted the quartic Higgs coupling gH4 against the cubic Higgs
coupling gH3. These are the coefficients of the terms (H1)4 and (H1)3, respectively, in the
Lagrangian; in the case of gH3 we have multiplied the coefficient of (H1)3 by gZZ /gZZ  in
order to take into account the possibility that the field H1 has the wrong sign. One sees
that the threeHiggs coupling may be almost twenty times larger than in the SM. Also,
that coupling may be zero or even negative, i.e. it may have a sign opposite to the one in the
(4.27)
(4.28)
(4.29)
mH1 = 125 GeV, higher than 150 GeV; green points have all those masses higher than 200 GeV,
and magenta points have all of them higher than 250 GeV.
for the colours is the same as in figure 4. The solid line and the dashed line are phenomenological
±
bounds extracted from figure 18 of ref. [22].
figure 4.
as in figure 4. The black cross indicates the values of gH4 and gH3 in the SM.
SM. The fourHiggs coupling is always larger than the corresponding SM coupling; it may
at most be 60% larger than in the SM. We point out that, in a general
twoHiggsdoubletmodel, the threeHiggs coupling has less freedom (it may at most be ten times larger than
in the SM) than in this model, while the fourHiggs coupling has much more freedom than
in this model — it may have values from zero until almost fifteen times larger than in the
SM [23]. Therefore, a measurement of gH3 — of the cubic interaction of the 125 GeV scalar
— may produce a large surprise and even distinguish this threeHiggsdoublet model from
the most general twoHiggsdoublet one.
In figure 9 we have plotted gZZ  against the quartic Higgs coupling and against the
cubic Higgs coupling. Notice that, although in our search we have restricted gZZ  to
have values in the range from 0.9 to 1, we have ended up obtaining only points with
gZZ  > 0.94. This is because we have restricted all the scalar masses (except the one
of H1) to be larger than 150 GeV; larger scalar masses require a larger gZZ  because the
values of gZZ  approach unity when the masses of all the new scalars are higher — this is
the decoupling limit.
5
Conclusions
In this paper we have constructed various extensions of the SM that yield predictions for
the effective lightneutrino Majorana mass matrix M given in terms of the chargedlepton
mass ratios. We have produced twelve models Mαpq, where α ∈ {e, µ, τ } and p, q ∈ {1, 2}.
Models Mα1q predict
Mββ
Mγγ
=
m2β
m2γ
,
Mαβ
Mαγ
=
mβ
mγ
(5.1)
the same as in figure 4. The dashed vertical lines indicate the values of gH3 and gH4 in the SM.
where α 6= β 6= γ 6= α, whereas models Mα2q predict
Furthermore, models Mαp1 predict whereas models Mαp2 predict
Mββ
Mγγ
=
m2γ
m2
β
,
Mαβ
Mαγ
=
mγ
mβ
.
arg hMγγ (Mαβ)2 Mβ∗β Mα∗γ 2i = 0,
arg hMβ∗βMγ∗γ (Mβγ )2i = 0,
arg Mα∗αMβ∗γ MαβMαγ = 0.
Mµ
Mττ
=
mµ2
mτ2
,
Meµ
Meτ
=
mµ
mτ
,
(5.2)
(5.3)
(5.4a)
(5.4b)
(5.5)
In practice, the conditions (5.3) or (5.4) are not so important; this is because
conditions (5.1) or (5.2) mean that two matrix elements of M are relatively small and lead
to our models being approximations to twotexturezero cases. Thus, eight of our twelve
models are able to correctly fit the data:
• Models Mµ 1q for q = 1, 2, which are approximations to case A1.
• Models Mτ1q for q = 1, 2, which are approximations to case A2.
• Models Me1q for q = 1, 2, which are approximations to case B3.
• Models Me2q for q = 1, 2, which are approximations to case B4.
The four models Mµ 2q and Mτ2q are not compatible with the phenomenological data and
are therefore excluded.
We have emphasized that our models Me1q lead, just from the two conditions
to a vast predictive power, viz. δ ≈ 3π/2, α21 ≈ 0, α31 ≈ π, and almost maximal
neutrinoless doublebeta decay for either a normal or an inverted neutrino mass spectrum.
Moreover, the quadrant of θ23 is correlated with the type of mass spectrum and θ23
approaches π/4 when the neutrino masses increase.
We have carefully worked out a scalar potential appropriate to our models Me1q. (With
slight modifications and no qualitatively different results, the potential is also appropriate
to models Mµ 1q and Mτ1q.) Our assumptions were the following:
• There are only three Higgs doublets φ1,2,3.
• There is an interchange symmetry φ2 ↔ φ3 that is not softly broken in the quadratic
part of the scalar potential.
HJEP03(218)5
• The potential has an unbroken symmetry under φ1 → −φ1.
• The vacuum expectation values are real.
• The symmetry φ1 → −φ1 is extended to the quark sector in such a way that only φ1
has Yukawa couplings to the quarks; the physical neutral scalars therefore have no
flavourchanging Yukawa couplings. CP violation is hard, i.e. it originates in complex
Yukawa couplings.
• The particle with mass 125 GeV discovered at LHC is the lightest physical scalar.
Through a careful simulation we have found the appropriate ranges for the various
parameters of the scalar potential. The physicalscalar masses cannot be much higher than a few
hundred GeV.
Acknowledgments
L.L. thanks Pedro M. Ferreira, Joa˜o Paulo Silva, and Igor Ivanov for useful discussions.
D.J. thanks the Lithuanian Academy of Sciences for support through the project DaFi2017.
The work of L.L. is supported by the Portuguese Funda¸ca˜o para a Ciˆencia e a Tecnologia
through the projects CERN/FISNUC/0010/2015 and UID/FIS/00777/2013, which are
partially funded by POCTI (FEDER), COMPETE, QREN, and the European Union.
A
Unitarity bounds for a 3HDM with Z2 × Z2 × Z2 symmetry
A.1
General case
We consider the most general threeHiggsdoublet model with Z(1)
2 × 2 × Z(23) symmetry,
Z(2)
where
Z(21) : φ1 → −φ1;
Z(22) : φ2 → −φ2;
Z(23) : φ3 → −φ3.
(A.1)
It is immaterial in this appendix whether any of the symmetries (A.1) is softly broken;
here we just deal with the quartic part of the potential
Vquartic = Λ1 φ1φ1
†
2
+ Λ2 φ2φ2
+ Λ3 φ3φ3
2
+ Λ4 φ†1φ1 φ†2φ2 + Λ5 φ†1φ1 φ†3φ3 + Λ6 φ†2φ2 φ†3φ3
where the letters a, . . . , f denote creation/destruction operators as well as the
corresponding particles. The (non)existence of vacuum expectation values is immaterial for the
unitarity bounds, therefore we neglect them in the notation (A.3). We denote the
Hermitianconjugate operators through bars: a† → a¯, b† → ¯b, and so on. Then,
a !
b
2
where Λ1,...,9 are real and Λ10,11,12 are in general complex. We follow ref. [24] to compute
HJEP03(218)5
the unitarity bounds on the parameters of the potential (A.2). For notational simplicity,
we write
(A.2a)
(A.2b)
(A.2c)
(A.2d)
(A.3)
(A.4a)
(A.4b)
(A.4c)
(A.4d)
(A.4e)
(A.4f)
(A.4g)
(A.4h)
(A.4i)
(A.4j)
(A.4k)
(A.4l)
(A.4m)
(A.4n)
(A.4o)
(A.5)
We must consider all the 2 → 2 scatterings that various pairs of particles may suffer among
themselves. For instance, the three states aa, cc, and ee may, at treelevel, scatter through
a matrix
Vquartic = Λ1 a¯a¯aa + ¯b¯bbb + 2a¯¯bab
+ Λ2 c¯c¯cc + d¯d¯dd + 2c¯d¯cd
+ Λ3 e¯e¯ee + f¯f¯f f + 2e¯f¯ef
+ Λ4 a¯c¯ac + ¯bd¯bd + a¯d¯ad + ¯bc¯bc
+ Λ5 a¯e¯ae + ¯bf¯bf + a¯f¯af + ¯be¯be
+ Λ6 c¯e¯ce + d¯f¯df + c¯f¯cf + d¯e¯de
+ Λ7 a¯c¯ac + ¯bd¯bd + a¯d¯bc + ¯bc¯ad
+ Λ8 a¯e¯ae + ¯bf¯bf + a¯f¯be + ¯be¯af
+ Λ9 c¯e¯ce + d¯f¯df + c¯f¯de + d¯e¯cf
+ Λ10 a¯a¯cc + ¯b¯bdd + 2a¯¯bcd
+ Λ1∗0 c¯c¯aa + d¯d¯bb + 2c¯d¯ab
+ Λ11 a¯a¯ee + ¯b¯bf f + 2a¯¯bef
+ Λ1∗1 e¯e¯aa + f¯f¯bb + 2e¯f¯ab
+ Λ12 c¯c¯ee + d¯d¯f f + 2c¯d¯ef
+ Λ1∗2 e¯e¯cc + f¯f¯dd + 2e¯f¯cd .
2Λ1 2Λ10 2Λ11
2Λ∗10 2Λ2 2Λ12 .
2Λ∗11 2Λ∗12 2Λ3
The scattering matrices of the states (ad, bc), (af, be), and (bc, de) are
Λ4 Λ7 !
Λ7 Λ4
,
Λ5 Λ8 !
Λ8 Λ5
,
Λ6 Λ9 !
Λ9 Λ6
,
respectively. The scattering matrices of the states ad¯, ¯bc , af¯, ¯be , and f¯c, d¯e are
(A.6)
(A.8)
(A.9)
(A.10a)
(A.10b)
(A.10c)
(A.11)
respectively. The scattering matrix of the states a¯b, cd¯, ef¯ is
The scattering matrix of the states a¯a, ¯bb, c¯c, d¯d, e¯e, f¯f is
2Λ1 Λ7 Λ8
Λ4 + Λ7
Λ5 + Λ8
2Λ1
4Λ1
Λ4
Λ5
Λ4 + Λ7
Λ5 + Λ8
Λ4 + Λ7
Λ4
4Λ2
2Λ2
Λ6
Λ6 + Λ9
Λ4
Λ4 + Λ7
2Λ2
4Λ2
Λ6
Λ6 + Λ9
Λ5 + Λ8
Λ6 + Λ9
Λ5
Λ6
4Λ3
2Λ3
Λ5
Λ6
2Λ3
4Λ3
Λ5 + Λ8
Λ6 + Λ9
.
In order to guarantee unitarity, we must enforce the condition that the moduli of all the
eigenvalues of these matrices (and of a few more analogous matrices) are smaller than 4π.
After some effort we find that those eigenvalues are
Λ4 ± Λ7,
Λ4 ± 2 Λ10 ,
Λ5 ± Λ8,
Λ5 ± 2 Λ11 ,
Λ6 ± Λ9,
Λ6 ± 2 Λ12 ,
Λ4 + 2Λ7 ± 6 Λ10 ,
Λ5 + 2Λ8 ± 6 Λ11 ,
Λ6 + 2Λ9 ± 6 Λ12 ,
and the eigenvalues of the matrices (A.5), (A.8), and
Λ4 2Λ10 !
2Λ∗10 Λ4
,
Λ5 2Λ11 !
2Λ∗11 Λ5
,
Λ6 2Λ12 !
2Λ∗12 Λ6
,
(A.7)
6Λ1
2Λ4 + Λ7 2Λ5 + Λ8
2Λ4 + Λ7
6Λ2
2Λ6 + Λ9 .
2Λ5 + Λ8 2Λ6 + Λ9
6Λ3
In our case there is an additional symmetry φ2 ↔ φ3 in the potential, and that simplifies
things much. Comparing equations (4.2) and (A.2), we see that
(A.12a)
(A.12b)
(A.12c)
(A.12d)
(A.12e)
(A.12f)
(A.12g)
(A.12h)
(A.13a)
(A.13b)
(A.13c)
(A.14)
(B.1a)
(B.1b)
The quantities (A.10) then become
λ3 ± λ5,
λ3 ± 2λ7,
and the matrices (A.5), (A.8), and (A.11) become
2λ1 2λ7 2λ7
2Λ7 2λ2 2λ8 ,
2λ7 2λ8 2λ2
2λ1 λ5 λ5
λ5 λ6 2λ2
λ5 2λ2 λ6 ,
6λ1
2λ3 + λ5 2λ3 + λ5
6λ2
The matrices (A.14) are 2–3 symmetric and therefore their eigenvalues are easy to find.
One thus obtains the quantities in the lefthand sides of inequalities (4.23).
B
Other stability points
Besides the vacuum state given by equations (4.6) and (4.7), there are several other stability
points of the potential. The vacuum state must have a lower value of the potential than
all other stability points. Therefore we must consider as many stability points as we can
and, for each of them, compute the expectation value of the potential. That is what we do
in the following.
1. Equations (4.5) have solutions with v1 6= 0 and v3 = ±v2. They are
v12 =
2 (λ2 + l4) µ 1 − 4l3 (µ 2 ± µ 3) ,
v22 = −2l3µ 1 + 2λ1 (µ 2 ± µ 3) .
8l32 − 4λ1 (λ2 + l4)
Plugging v3 = ±v2 together with equations (B.1) into equation (4.3), one obtains
V0 = V0(1±)
2. The point v1 = v2 = v3 = 0 has
3. If v2 = v3 = 0 but v1 6= 0, there is a stability point with
yielding
to be positive and
V0 = V0(2)
≡ 0.
,
V0 = V0(3)
≡ − 4λ1
4. If v1 = 0 but v2 6= 0 and v3 6= 0, we may analytically entertain the possibility that
the VEVs of φ02 and φ03 have a relative phase ϑ. We take in this case both v2 and v3
V0 = µ 2 v22 + v32 + λ2 v24 + v34 + (λ4 + λ6) v22v32
+ 2µ 3v2v3 cos ϑ + 2λ8v22v32 cos (2ϑ).
The stationarity equations are
0 = µ 3 sin ϑ + 2λ8v2v3 sin (2ϑ),
0 = µ 2v2 + µ 3v3 cos ϑ + 2λ2v23 + (λ4 + λ6) v2v32 + 2λ8v2v32 cos (2ϑ),
0 = µ 2v3 + µ 3v2 cos ϑ + 2λ2v33 + (λ4 + λ6) v22v3 + 2λ8v2v3 cos (2ϑ).
2
This leads to the following possibilities:
(a) cos ϑ = ±1 and v3 6= v2. Then,
(B.2)
(B.3)
(B.4)
(B.5)
(B.6)
(B.7)
(B.8a)
(B.8b)
(B.9)
(B.10)
(B.11)
Plugging cos ϑ = ±1 and equations (B.8) into equation (B.6), one obtains
(b) cos ϑ = ±1 and v3 = v2. One then has
leading to
v22 + v32 = − 2λ2
,
µ 2
,
µ 2 ± µ 3
V0 = V0(5±)
(µ 2 ± µ 3)
2
≡ − 2λ2 + λ4 + λ6 + 2λ8
and to
Of course, this stability point only exists if cos ϑ ≤ 1, viz.
4λ8µ 2
1 ≤ (2λ2 + λ4 + λ6 − 2λ8) µ 3
.
(B.12)
(B.13)
(B.14)
Attribution License (CCBY 4.0), which permits any use, distribution and reproduction in
any medium, provided the original author(s) and source are credited.
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