Table Of ContentDynamical seesaw mechanism for Dirac neutrinos
Jos´e W.F. Valle1, and C.A. Vaquera-Araujo1,
∗ †
1AHEP Group, Institut de F´ısica Corpuscular – C.S.I.C./Universitat de Val`encia, Parc Cient´ıfic de Paterna.
C/ Catedra´tico Jos´e Beltr´an, 2 E-46980 Paterna (Valencia) - SPAIN
(Dated: January 21, 2016)
Sofarwehavenotbeenableto establish that, astheoreticallyexpected, neutrinosaretheirown
anti-particles. Here we propose a dynamical way to account for the Dirac nature of neutrinos and
the smallness of their mass in terms of a new variant of the seesaw paradigm in which the energy
scale of neutrino mass generation could be accessible to the current LHC experiments.
PACSnumbers: 14.60.Pq,12.60.Cn,14.60.St
6
1
0
2 INTRODUCTION
n
a Although neutrinos are not proven experimentally to be Dirac or Majorana fermions, the leading theoretical ex-
J
pectation is that they are Majorana. Indeed there is a widespread paradigm that ascribes the smallness of neutrino
0
2 massesrelativetotheotherStandardModelfermionmassestotheirchargeneutrality. Thisisnaturallyincorporated
by assuming that neutrinos acquire Majorana masses from a lepton number violating operator, such as Weinberg’s
]
h
dimension five operator or similar higher order ones. Indeed conventional type-I [1–5] or type-II [4–6] formulations of
p
- the seesaw mechanism lead typically to Majorana neutrinos, irrespective of whether the seesaw is realized at high or
p
e atlowmassscale,inthespiritofthemodelsconsideredin[7]. Untiltheobservationofneutrinolessdoublebetadecay
h
[8] becomes unambiguously confirmed [9] the possibility remains that neutrinos can be Dirac particles after all.
[
The theoretical challenge to account for this possibility is then twofold: i) to predict Dirac neutrinos, and
1
ii), to understand dynamically their small mass. Regarding the first we need to use extra symmetries beyond
v
7 SU(3) ⊗SU(2) ⊗U(1) gauge symmetry, otherwise massive neutrinos are generally expected to be Majorana par-
c L Y
3
ticles [4]. To this end, within the standard SU(3) ⊗SU(2) ⊗U(1) electroweak gauge structure, one may impose a
2 c L Y
5 conserved lepton number, so as to obtain Dirac neutrinos. If the lepton number assignment is non-standard one may
0
obtain naturally light Dirac neutrinos from calculable radiative corrections [10]. Likewise, one may consider schemes
.
1
basedonflavorsymmetries,assuggestedin[11]. Unfortunatelyinthesimplestrealizationofthisideathesmallnessof
0
6 neutrino mass is put in by hand. Another approach would be to appeal to the existence of extra dimensions [12, 13].
1
Alternativelyonemayextendthegaugegroupitself,soasto(atleastpartially)includetheleptonnumbersymmetry,
:
v
for example, by using the extended SU(3) ⊗SU(3) ⊗U(1) gauge structure [14] which predicts the number of
i c L X
X
fermion generations to match the number of colors. Although modern SU(3) ⊗SU(3) ⊗U(1) schemes have
c L X
r
a Majorana neutrinos, with either radiatively induced [15, 16] or seesaw–type masses [17, 18], the simplest original
formulation predicts Dirac neutrino masses [19]. Apart from not being able to account for current oscillation data,
that original formulation did not address the question of how to account for the observed smallness of neutrino mass.
In this letter we focus on the possibility of having naturally light Dirac neutrinos with masses induced a la see-
saw. We adopt the SU(3) ⊗SU(3) ⊗U(1) gauge structure because of its unique features, with respect to other
c L X
electroweak extensions based, for example, on left-right symmetry. Indeed, our new variant of the seesaw mechanism
for Dirac neutrinos makes use of the peculiar features of the SU(3) ⊗SU(3) ⊗U(1) based models. The resulting
c L X
scheme provides a Dirac seesaw alternative to the approaches considered in [15, 16] and [17, 18].
2
THE MODEL
Our starting point is a variant of the SU(3) ⊗SU(3) ⊗U(1) gauge framework with the anomaly–free matter
c L X
content given in Tab. (I). Notice that one has the same set of “gauge–charged” fields as in [15], for example the
left-handed leptons transform as
(cid:96)
−
ψ(cid:96) = ν , (1)
L (cid:96)
N
(cid:96)
L
with (cid:96) = 1,2,3 ≡ e, µ, τ. In contrast to what is assumed in [15], however, we add more gauge singlet leptons and
we change the transformation properties of two of the scalar multiplets under L, the ungauged piece of the lepton
number symmetry. Indeed, in this model the electric charge can be written in terms of the U(1) generator X and
X
the diagonal generators of the SU(3) , whereas lepton number has a gauge component as well as a complementary
L
global one:
1
Q = T + √ T +X, (2)
3 8
3
4
L = √ T +L. (3)
8
3
ψ(cid:96) (cid:96) S(cid:96) ,S˜(cid:96) Q1,2 Q3 uˆ dˆ φ φ φ
L R R R L L R R 0 1 2
SU(3) 1 1 1 3 3 3 3 1 1 1
c
SU(3) 3∗ 1 1 3 3∗ 1 1 3∗ 3∗ 3∗
L
U(1) −1 −1 0 0 +1 +2 −1 +2 −1 −1
X 3 3 3 3 3 3 3
L −1 −1 1 −2 +2 0 0 +2 −4 −4
3 3 3 3 3 3
Zaux ω ω ω ω2 ω2 ω2 ω2 1 1 1
3
TABLE I: Matter content of the model, where uˆ ≡(u ,c ,t ,U ) and dˆ ≡(d ,s ,b ,D ,D(cid:48) ).
R R R R R R R R R R R
Let us now explain the new features of the model in relation to previous variants of the SU(3) ⊗SU(3) ⊗U(1)
c L X
electroweak model.
Change in the matter sector: In addition to the two-component neutral fermions N needed to fill up the SU(3)
L L
anti-triplets, in our new model we introduce two sequential sets of L–carrying gauge singlet leptons denoted as S ,
R
S˜ .
R
Change in the scalar sector: As before, three scalar anti-triplets φ ∼ (3 ,+2/3) and φ ∼ (3 ,−1/3) are
0 ∗ 1,2 ∗
responsibleforthespontaneousbreakdownoftheextendedelectroweakSU(3) gaugesymmetry. However,incontrast
L
to the formulation presented in [15], in the present framework the scalar triplets φ and φ have the same L charge.
1 2
Following the notation of [19] the most general pattern for the vacuum expectation values (VEVs) of the fields is
k 0 0
0
1 1 1
(cid:104)φ0(cid:105)= √2 0 , (cid:104)φ1(cid:105)= √2 k1 , (cid:104)φ2(cid:105)= √2 k2 , (4)
0 n n
1 2
3
where the isosinglet VEVs n and n characterize the SU(3) breaking scale. Correspondingly k , k and k are the
1 2 L 0 1 2
VEVs of the SU(2) ⊂SU(3) doublets, so we expect that k ,k ,k (cid:28)n ,n .
L L 0 1 2 1 2
In the fermion sector, the relevant Yukawa terms invariant under the gauge symmetry and the auxiliary Zaux are
3
−L = y(cid:96)ψ¯ l φ +y ψ¯ S φ +y˜ ψ¯ S φ +y ψ¯ S˜ φ +y˜ ψ¯ S˜ φ
f L R 0 1 L R 1 1 L R 2 2 L R 1 2 L R 2
+ yuQ¯1,2uˆ φ +yˆuQ¯3 uˆ φ +y˜uQ¯3 uˆ φ
L R ∗0 L R 1 L R 2
+ ydQ¯3 dˆ φ +yˆdQ¯1,2dˆ φ +y˜dQ¯1,2dˆ φ +h.c., (5)
L R 0 L R ∗1 L R ∗2
where contraction of the flavor indices is implicitly assumed. An important feature of the model is the fact that the
L symmetry is preserved in the lepton sector. The role of the discrete symmetry Zaux is to forbid the gauge invariant
3
ψTC 1ψ φ term in order to simplify the following discussion.
L − L 0
Concerning the symmetry breaking sector, the most general CP conserving scalar potential compatible with the
SU(3) ⊗SU(3) ⊗U(1) gauge symmetry as well as L invariance is given by
c L X
V =(cid:88)(cid:16)µ2i |φi|2+λi|φi|4(cid:17)+(cid:88)λij|φi|2|φj|2+(cid:88)λ˜ij(φ†iφj)(φ†jφi)
L
i i=j i=j
(cid:54) (cid:54)
(cid:16) (cid:17) (cid:88) (cid:16) (cid:17)
+µ2s φ†1φ2+h.c. + λ(cid:48)i φ†1φ2+h.c. |φi|2 (6)
i
(cid:104) (cid:105) (cid:104) (cid:105)
+λ (φ†1φ2)(φ†1φ2)+h.c. +λ˜ (φ†1φ0)(φ†0φ2)+h.c. ,
with i = 0,1,2. The key ingredient of our present construction is the inclusion of a L violating piece in the scalar
potential:
V =f(φ φ φ +h.c.) , (7)
0 1 2
(cid:54)L
where the term φ φ φ is understood as the fully antisymmetric product of the scalar field components. The mass
0 1 2
dimension one parameter f is expected to be small, in the sense that the L symmetry is restored in the scalar sector
in the limit f → 0 1. This symmetry can be interpreted as a chiral symmetry which will control the massiveness of
neutrinos, as we show in the next section.
For simplicity, all VEVs and Yukawa coefficients are assumed to be real in the following analysis. From the
minimization conditions of the total potential V = V +V : ∂V/∂k = ∂V/∂n = ∂V/∂n = ∂V/∂k = 0, the
0 1 2 2
L (cid:54)L
1 Thescalarpotentialinthislimithasbeenstudiedin[20].
4
dimensionful parameters µ and f are determined as
0,1,2
(cid:40) (cid:2) (cid:0) (cid:1) (cid:0) (cid:1)(cid:3)(cid:41)
µ2 =− 1 2λ k2+λ (cid:0)k2+n2(cid:1)+λ (cid:0)k2+n2(cid:1)+ λ(cid:48)0 k12 2k22+n22 +2k1k2n1n2+n21 k22+2n22
0 2 0 0 01 1 1 02 2 2 k k +n n
1 2 1 2
(cid:34) (cid:0) (cid:1) (cid:0) (cid:1)(cid:35)
(k n −k n )2 2µ2+λ k2+n2 +λ k2+n2
− 2 1 1 2 λ˜ +2λ+ s (cid:48)1 1 1 (cid:48)2 2 2 ,
2k2 12 k k +n n
0 1 2 1 2
(cid:40) (cid:0) (cid:1)(cid:2) (cid:0) (cid:1) (cid:3)(cid:41)
µ2 =− 1 λ k2+2λ (cid:0)k2+n2(cid:1)+(2λ+λ +λ˜ )(cid:0)k2+n2(cid:1)+ k22+n22 λ(cid:48)0k02+λ(cid:48)2 k22+n22 +2µ2s
1 2 01 0 1 1 1 12 12 2 2 k k +n n
1 2 1 2
(cid:2) (cid:0) (cid:1) (cid:0) (cid:1)(cid:3)
λ k2 3k2+n2 +4k k n n +n2 k2+3n2
− (cid:48)1 1 2 2 1 2 1 2 1 2 2 , (8)
2(k k +n n )
1 2 1 2
(cid:40) (cid:0) (cid:1)(cid:2) (cid:0) (cid:1) (cid:3)(cid:41)
µ2 =− 1 λ k2+2λ (cid:0)k2+n2(cid:1)+(2λ+λ +λ˜ )(cid:0)k2+n2(cid:1)+ k12+n21 λ(cid:48)0k02+λ(cid:48)1 k12+n21 +2µ2s
2 2 02 0 2 2 2 12 12 1 1 k k +n n
1 2 1 2
(cid:2) (cid:0) (cid:1) (cid:0) (cid:1)(cid:3)
λ k2 3k2+n2 +4k k n n +n2 k2+3n2
− (cid:48)2 1 2 2 1 2 1 2 1 2 2 ,
2(k k +n n )
1 2 1 2
(cid:40)2λ+λ˜ λ k2+λ (k2+n2)+λ (cid:0)k2+n2(cid:1)+2µ2(cid:41)
f =(k n −k n ) √ 12 + (cid:48)0 0 (cid:48)1 √1 1 (cid:48)2 2 2 s ,
1 2 2 1
2k 2k (k k +n n )
0 0 1 2 1 2
withtheremainingcondition∂V/∂k =0automaticallysatisfied. Weconcludethissectionpointingouttheinteresting
1
dependence of f on the VEV combination (k n −k n ). This means that we can interpret the last relation of Eq.(8)
1 2 2 1
as a statement that a small parameter f dynamically induces a small non-zero value for (k n −k n ).
1 2 2 1
NEUTRINO MASSES
After spontaneous symmetry breaking, the Dirac neutrino mass matrix becomes
−Lmass = √12(cid:16)ν¯L N¯L (cid:17)yy1nk1++yy˜˜1kn2 yy2nk1++yy˜˜2kn2 SS˜R +h.c., (9)
1 1 1 2 2 1 2 2 R
where y and y˜ are 3×3 Yukawa matrices. The light neutrino masses can be readily estimated in the one family
1,2 1,2
approximation, in which the neutrino mass matrix is diagonalized by a bi-unitary transformation M =U MU ,
diag ν† S
with
cosθ sinθ
α α
Uα ≈ , α=ν, S, (10)
−sinθ cosθ
α α
and
(cid:2) (cid:0) (cid:1) (cid:0) (cid:1)(cid:3)
2 k n y2+y2 +(k n +k n )(y˜ y +y˜ y )+k n y˜2+y˜2
tan2θ =− 1 1 1 2 1 2 2 1 1 1 2 2 2 2 1 2 ,
ν (k2−n2)(y2+y2)+2(k k −n n )(y˜ y +y˜ y )+(k2−n2)(y˜2+y˜2)
1 1 1 2 1 2 1 2 1 1 2 2 2 2 1 2 (11)
(cid:2)(cid:0) (cid:1) (cid:3)
2 k2+n2 y y +(k k +n n )(y˜ y +y˜ y )+(k2+n2)y˜ y˜
tan2θ =− 1 1 1 2 1 2 1 2 1 2 2 1 2 2 1 2 ,
S (k2+n2)(y2−y2)+2(k k +n n )(y˜ y −y˜ y )+(k2+n2)(y˜2−y˜2)
1 1 1 2 1 2 1 2 1 1 2 2 2 2 1 2
yielding eigenstates with masses
(cid:114)
1 (cid:16) (cid:112) (cid:17)
m = A∓ A2−4B2 ,
∓ 2
A=(cid:0)k2+n2(cid:1)(cid:0)y2+y2(cid:1)+2(k k +n n )(y˜ y +y˜ y )+(k2+n2)(cid:0)y˜2+y˜2(cid:1) , (12)
1 1 1 2 1 2 1 2 1 1 2 2 2 2 1 2
B =(k n −k n )(y y˜ −y y˜ ).
1 2 2 1 1 2 2 1
5
Thus, up to corrections of O(B2/A3/2), the resulting masses for the light and heavy neutrinos are
|(k n −k n )(y y˜ −y y˜ )|
m ≈ 1 2 2 1 1 2 2 1 ,
light (cid:112)
2[(k2+n2)(y2+y2)+2(k k +n n )(y˜ y +y˜ y )+(k2+n2)(y˜2+y˜2)]
1 1 1 2 1 2 1 2 1 1 2 2 2 2 1 2 (13)
(cid:114)
1
m ≈ [(k2+n2)(y2+y2)+2(k k +n n )(y˜ y +y˜ y )+(k2+n2)(y˜2+y˜2)].
heavy 2 1 1 1 2 1 2 1 2 1 1 2 2 2 2 1 2
φ φ
0 2,1
h i h i
f
φ
1,2
SR,S˜R ν¯L,N¯L
FIG. 1: Type–II–like dynamical seesaw mechanism for Dirac neutrino mass.
The mixing angle θ is small by virtue of the VEV hierarchy k ,k ,k (cid:28) n ,n 2. For large n ,n VEVs one
ν 0 1 2 1 2 1 2
clearlyobtainsthestandardseesawbehavior,inwhichthesmallneutrinomassemergesfromtheparametersgoverning
the scale characterizing the mass of the messenger particle, in this case a heavy scalar boson, see Fig. 1.
However Eq.(13) contains further crucial information, namely the fact that the light neutrino mass is determined
by the scale (k n −k n ), the same combination of VEVs found in the last relation in Eq.(8). The smallness of the
1 2 2 1
neutrino mass can then be understood as a consequence of the interplay between the f term in the scalar potential
and V . The presence of a small f parameter (quantifying the amount of L violation in the scalar sector) and large
L
quartic couplings enforces a nearly parallel dynamical alignment for the φ and φ VEVs, which in turn can lead to
2 3
a tiny mass for the active neutrino even without imposing a large hierarchy among the k and n scales.
1,2 1,2
Written in terms of f, and using the approximation n ∼ n (cid:29) k ∼ k,k ,|µ |, the mass of the light neutrino
1,2 1,2 0 s
simplifies to
|fk (y y˜ −y y˜ )|
m ≈ 0 1 2 2 1 . (14)
light (cid:113)
|λ˜ +λ +λ +2λ|n (y +y˜ )2+(y +y˜ )2
12 (cid:48)1 (cid:48)2 1 1 2 2
Assuming|λ˜ +λ +λ +2λ|∼O(1)intheaboveequation,wecanseethattheresultingmassispotentiallysuppressed
12 (cid:48)1 (cid:48)2
bythreedifferentsources: (i)thefactork /n;(ii)thesmallscalef associatedtotheL–symmetryprotection,and(iii)
0
thedeterminant-likeYukawacombination(y y˜ −y y˜ ). Weconcludethissectionwithanillustrativeexampleofhow
1 2 2 1
these three sources can act in synergy: setting f ∼O(1)keV, k ∼O(102)GeV, n∼O(10)TeV, and y =y˜ =y+δ,
0 1 2
y = y˜ = y −δ with δ ∼ O(10 2), a naturally small neutrino mass m ∼ O(10 1)eV is obtained without the
2 1 − light −
need to invoke superheavy physics. The above estimate of light neutrino makes use of the one family approximation.
However, generalization to three families is straightforward, using the perturbative block-diagonalization technique
developed in [5].
2 NoticethatθS becomesmaximalinthelimity1→y2,y˜1→y˜2.
6
DISCUSSION AND CONCLUSIONS
Summarizing, inthisletterwehavepresentedanovelmechanismforDiracneutrinomassgenerationinthecontext
of SU(3) ⊗SU(3) ⊗U(1) based models. The light neutrino mass is induced at tree-level a la seesaw as indicated
c L X
in Fig. 1. Its smallness is guaranteed by three independent features:
1. ThesuppressionimposedbytheratiooftheSU(2) andSU(3) breakingscales, whichbyitselfcanaccountfor
L L
the small neutrino mass by invoking a large hierarchy among these scales as in the standard high-scale seesaw
mechanism.
2. ThedynamicalalignmentoftheVEVs, inducedbythepresenceofasmalltrilinearterminthescalarpotential,
which is ultimately related to the smallness of L violation encoded by the characteristic scale f. The presence
of this key ingredient makes it possible to have a low-scale realization of our seesaw mechanism, at an energy
scale accessible to the current LHC experiments.
3. TheadditionalsuppressionprovidedbythepeculiardependenceoftheneutrinomassontheYukawacoefficients
ofthemodel,whichfavorsasmallneutrinomassinthecaseofaYukawaalignmentanalogoustotheonedisplayed
by the VEVs. Potentially this might have a dynamical origin say, in string theories.
Before closing let us note that a low-scale seesaw realization in the present context would bring in a very rich
phenomenology with new quarks, new gauge bosons as well as new scalars, all of them lurking within the range that
can be explored in flavor studies and the LHC. Its detailed study lies beyond the scope of this letter. In particular
the model can fit the recent hint for a di-photon resonance in a natural way [21, 22].
Acknowledgements
This work is supported by the Spanish grants FPA2014-58183-P, Multidark CSD2009-00064, SEV-2014-0398
(MINECO) and PROMETEOII/2014/084 (Generalitat Valenciana). C.A.V-A. acknowledges support from CONA-
CyT (Mexico), grant 251357.
∗ Electronic address: valle@ific.uv.es
† Electronic address: vaquera@ific.uv.es
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