A model of neutrino mass and dark matter with large neutrinoless double beta decay

0.8 MB·

by  Julien Alcaide

FTUV-17-0104, IFIC-16-97
A model of neutrino mass and dark matter
with large neutrinoless double beta decay
Julien Alcaide1, Dipankar Das2, Arcadi Santamaria3
Departament de F´ısica T`eorica, Universitat de Val`encia and IFIC, Universitat de Val`encia-CSIC
Dr. Moliner 50, E-46100 Burjassot (Val`encia), Spain
7
1
0 Abstract
2
Weproposeamodelwhereneutrinomassesaregeneratedatthreelooporderbutneutrinolessdoublebeta
n
a decay occurs at one loop. Thus we can have large neutrinoless double beta decay observable in the future
J experiments even when the neutrino masses are very small. The model receives strong constraints from the

5 neutrinodataandleptonflavorviolatingdecays,whichsubstantiallyreducesthenumberoffreeparameters.
Our model also opens up the possibility of having several new scalars below the TeV regime, which can

] be explored at the collider experiments. Additionally, our model also has an unbroken Z symmetry which
2
h
allows us to identify a viable Dark Matter candidate.
p
-
p
e
h
[ 1 Introduction

1
v
2 Almost all the extensions of the Standard Model (SM) directed towards an explanation for the neutrino masses
0 brings in the possibility of lepton number violation (LNV) as an outcome. It is well known that neutrinoless
4 double beta decay (0νββ decay) which is a convincing signature for LNV, will be an inevitable consequence if
1
the neutrino has Majorana mass. All the experiments that are currently looking for 0νββ decay, are working
0
under the assumption that 0νββ decay proceeds through the Majorana neutrino propagator at the leading
.
1 order. This means, depending on the spectrum of the neutrino masses, the expected rate for 0νββ decay might
0
be too small to be observed in the experiments. But there exist scenarios where the mechanism for leading
7
order 0νββ decay is not directly correlated to the Majorana masses of the neutrinos. In such cases we can have
1
: thepossibilityoflarge0νββ decay evenwhentheneutrinoMajoranamassesaresmall. Manystudieshavebeen
v
performedinthisdirectioninthepast(seeRefs.[1–3]forageneraloverview,Refs.[4–22]forspecificmodelsand
i
X Refs. [23–26] for effective field theory (EFT) approaches). In Ref. [26], the authors performed an EFT analysis
r of the different ways of generating 0νββ decay and light neutrino masses by including operators involving only
a
leptons, Higgs and gauge bosons. This led to a class of interesting models where 0νββ decay was generated at
tree level whereas neutrino masses would appear only at two-loops (see Refs. [27] for example models in this
category).
The model in Ref. [27] contains an SU(2) singlet doubly charged scalar like in the Zee-Babu model [28–30],
L
an SU(2) triplet scalar with hypercharge +1 and a real singlet scalar. A Z symmetry, which is later broken
L 2
spontaneously, is required to prevent tree-level neutrino masses. The model is economical in the sense that it
contains no new fermions and by design, it gives new contributions to 0νββ decay, which, in principle, can be
large. Additionally,ithasarichphenomenologywhichcanbeprobedthroughthesearchesfortheleptonflavor
violating (LFV) signals and/or the direct searches for the new scalars in the collider experiments.
[email protected]
[email protected]
[email protected]
1
In this article we will present a simple variation of the model in Ref. [27]. Our new model will have the same
field content as in Ref. [27], except that the Z symmetry will not be broken spontaneously. Consequently,
2
0νββ decay will now occur at one-loop whereas neutrino masses will appear at three-loop order. The fact the
Z is exact makes the model simpler and allows for a viable Dark Matter (DM) candidate: the lightest of the
2
electrically neutral Z -odd particles. On the other hand, the model keeps all the virtues of the previous model:
2
verypredictiveneutrinomassmatrix, large0νββ decay decay, richleptonflavourviolationphenomenologyand
new scalars which are in the sub-TeV region and therefore, are within the reach of the collider experiments in
the near future.
Our paper will be organized as follows. In Sec. 2 we lay out the scalar field content and the physical spectrum
of our model. In Sec. 3 we discuss the 0νββ decay and the bounds that follow from it. Neutrino masses and
constraints from LFV decays are discussed in Sec. 4 and Sec. 5 respectively. We analyze the feasibility of DM
in Sec. 6. Finally, we summarize our findings in Sec. 7.
2 The model
The scalar sector of our model contains the following fields:
(cid:26) (cid:27)
1
Φ= 2, ; χ= 3,1 ; κ++ = 1,2 ; σ =real singlet, (1)
2 { } { }
where,thenumbersinsidethecurlybracketsassociatedwiththefieldsrepresenttheirtransformationsproperties
under SU(2) and U(1) respectively. The normalization for the hypercharge is such that the electric charges
L Y
ofthecomponentfieldsaregivenby,Q=T +Y. Thefields,χandσ areoddunderanadditionalZ symmetry
3 2
which has been introduced to prevent the occurrence of tree-level neutrino masses as well as to ensure the
stability of the DM particle. The most general scalar potential involving these fields is given below:
V = m2 (cid:0)Φ†Φ(cid:1)+m2 Tr(cid:0)χ†χ(cid:1)+m2 κ2+ m2σσ2+λ (cid:0)Φ†Φ(cid:1)2+λ (cid:8)Tr(cid:0)χ†χ(cid:1)(cid:9)2
− Φ χ κ| | 2 Φ χ
+λ(cid:48) Tr(cid:104)(cid:0)χ†χ(cid:1)2(cid:105)+λ κ4+λ σ 4+λ (cid:0)Φ†Φ(cid:1)Tr(cid:0)χ†χ(cid:1)+λ(cid:48) (cid:0)Φ†χχ†Φ(cid:1)
χ κ| | σ| | Φχ Φχ
+λ (cid:0)Φ†Φ(cid:1) κ2+λ (cid:0)Φ†Φ(cid:1)σ2+λ κ2Tr(cid:0)χ†χ(cid:1)+λ σ2Tr(cid:0)χ†χ(cid:1)
Φκ Φσ κχ σχ
| | | |
+λσκ κ2σ2+(cid:110)µκκ++Tr(cid:0)χ†χ†(cid:1)+λ6σΦ†χΦ(cid:101) +h.c.(cid:111) , (2)
| |
where ‘Tr’ represents the trace over 2 2 matrices and Φ(cid:101) =iσ2Φ∗, with σ2 being the second Pauli matrix. We
×
can take all the parameters in the potential to be real without any loss of generality.
For the leptonic Yukawa sector, we have the following Lagrangian:
L = (L ) (Y ) ((cid:96) ) Φ+f (cid:96)TC−1((cid:96) ) κ+++h.c., (3)
Y − L a e ab R b ab a R b
where,L =(ν , (cid:96))T denotestheleft-handedleptondoubletand(cid:96) representstheright-handedchargedlepton
L (cid:96) L R
singlet. C isthechargeconjugationoperator. Wechoosetoworkinthemassbasisofthechargedleptonswhich
means,Y isadiagonalmatrixwithpositiveentriesandf isacomplexsymmetricmatrixwiththreeunphysical
e
phases.
2.1 The scalar spectrum
We do not want to break the Z symmetry spontaneously. Denoting by v the vacuum expectation values (vev)
2
of the doublet the minimization conditions read
m2 = λ v2. (4)
Φ Φ
2
Z -even particles Z -odd particles
2 2
SM fermions and gauge bosons, h and κ±± S, A, H, χ±, χ±±
Table 1: Z parity assignments to the physical particles in our model.
2
After spontaneous symmetry breaking (SSB) we represent the doublet and the triplet as follows:
(cid:18) (cid:19) (cid:18) (cid:19)
1 √2ω+ 1 χ+ √2χ++
Φ= , χ= , (5)
√2 v+h+iζ √2 ht+iA χ+
−
where, ω and ζ represent the Goldstones associated with the W and Z bosons respectively. Because of the
unbroken Z symmetry, only h and σ can have nontrivial mixing. This leads to a very simple scalar spectrum
2 t
as described below.
The masses for the doubly charged particles are given by,
1 1
m2 =m2 + λ v2, m2 =m2 + λ v2. (6)
κ++ κ 2 Φκ χ++ χ 2 Φχ
The mass of the singly charged scalar is given by,
1
m2 =m2 + (2λ +λ(cid:48) )v2. (7)
χ+ χ 4 Φχ Φχ
The pseudoscalar mass is given by,
1
m2 =m2 + (λ +λ(cid:48) )v2. (8)
A χ 2 Φχ Φχ
From Eqs. (6), (7) and (8) it is easy to see that the following correlation holds:
1
m2 m2 =m2 m2 = λ(cid:48) v2. (9)
χ+− χ++ A− χ+ 4 Φχ
In the CP even sector, the SM-like Higgs arises purely from the doublet, Φ, with mass m2 = 2λ v2. For the
h Φ
other two Z -odd scalars, we obtain the following mass matrix:
2
(cid:18) (cid:19)(cid:18) (cid:19)
VmSass = 12(cid:0)σ ht(cid:1) AB −CB hσt with, (10)
−
1 1
A=m2 +λ v2, B = λ v2, C =m2 + (λ +λ(cid:48) )v2. (11)
σ Φσ −√2 6 χ 2 Φχ Φχ
This mass matrix can be diagonalized by the following orthogonal rotation:
(cid:18) (cid:19) (cid:18) (cid:19)(cid:18) (cid:19)
S cosα sinα σ
= − , (12a)
H sinα cosα h
t
1(cid:110) (cid:112) (cid:111)
with, m2 = (A+C) (A C)2+4B2 , (12b)
H,S 2 ± −
2B
and, tan2α = , (12c)
A C
−
where we have implicitly assumed that ‘S’ is the lighter mass eigenstate. One can easily find the following
relations:
A = m2 sin2α+m2 cos2α, (13a)
H S
3
C = m2 cos2α+m2 sin2α=m2 , (13b)
H S A
B = sinαcosα(m2 m2), (13c)
− H − S
which imply,
m <m <m . (14)
S A H
Combining Eqs. (11) and (13c) we can express λ in terms of the physical parameter as follows:
6
λ = √2sinαcosα(cid:0)m2 m2(cid:1) . (15)
6 v2 H − S
The splittings between different scalar masses can be constrained further from the electroweak T-parameter.
The expression for the new physics contribution to the T-parameter is given by
(cid:20)
1 1
∆T = F(m2 ,m2 )+ F(m2 ,m2)
4πsin2θ M2 χ++ χ+ 2 χ+ A
W W
1 (cid:110) (cid:111) 1 (cid:110) (cid:111)(cid:21)
+ cos2α F(m2 ,m2 ) 2F(m2,m2 ) + sin2α F(m2 ,m2) 2F(m2,m2) , (16)
2 χ+ H − A H 2 χ+ S − A S
where, θ and M are the weak mixing angle and the W-boson mass respectively. The function, F(m2,m2),
W W 1 2
is given by,
1 (cid:90) d4k (cid:18) 1 1 (cid:19)2 m2+m2 m2m2 (cid:18)m2(cid:19)
F(m2,m2) 16π2 k2 = 1 2 1 2 log 1 . (17)
1 2 ≡ 2 (2π)4 k2+m2 − k2+m2 2 − m2 m2 m2
1 2 1− 2 2
Taking the new physics contribution to the T-parameter as [31]
∆T =0.05 0.12, (18)
±
we will require our model value of the T-parameter to be within the 2σ uncertainty range.
Inpassing,combiningEqs.(9)and(14),wenotethattwotypesofscalarmasshierarchiesarepossibledepending
on the sign of λ(cid:48) ,
Φχ
m >m >m >m >m , (19a)
H A χ+ χ++ S
or, m >m >m >m and m >m . (19b)
χ++ χ+ A S H A
In both cases, m can be arbitrary in principle.
κ++
3 Estimation of 0νββ decay
For new scalar masses of (1 TeV), the Majorana mass matrix element, M , will be very small (see Sec. 4 for
ee
O
details). As a result, the usual neutrino exchange diagram will contribute negligibly to 0νββ decay. The main
contribution to the 0νββ decay amplitude has been displayed in Fig. 1. From the diagram in Fig. 1 we can
easily estimate the effective e¯ec(u¯d)2 interaction giving rise to 0νββ decay
f∗ µ λ2
=2 ee κ 6 I (u γµd )(u γ d )e ec , (20)
L0νββ 16π2m2 m4 β L L L µ L R R
κ++ A
whereI isadimensionlessfunctionofthescalarmassesrunningintheloopwhichisexpectedtobe (1). We
β
O
have chosen the common scale of the loop to be the mass of the pseudoscalar part from the scalar triplet, m .
A
Of course the diagram in Fig. 1 is only one of the contributions in the mass insertion approach which allows
4
d Φ Φ Φ Φ
h i h ih i h i
e
W χ0 σ χ0 R
u κ
−−
u
e
R
W
χ
−−
d
Figure 1: One-loop diagram, in the mass insertion approach, contributing to neutrinoless double beta
decay.
us to give an estimate. A complete calculation of the function I in the physical basis has been presented in
β
Appendix A.
This type of interaction has been considered in the literature [32,33], where it was parametrized as follows:
G2
= F (cid:15) (u¯γµ(1 γ )d)(u¯γ (1 γ )d)e¯(1 γ )ec. (21)
L0νββ 2m 3 − 5 µ − 5 − 5
p
Comparing Eqs. (20) and (21) we obtain,
m f∗ µ λ2
(cid:15) = p ee κ 6 I . (22)
3 2G2 16π2m2 m4 β
F κ++ A
In Ref. [33], to set bounds on (cid:15) , the authors used the limits on the half-life for the 0νββ decay from the
3
most sensitive experiments of that time, namely, T0νββ(76Ge)>1.9 1025 yrs (HM [34]) and T0νββ(136Xe)>
1/2 × 1/2
1.6 1025 yrs (EXO-200 [35]). However KamLAND-Zen has recently obtained a stronger limit on the lifetime
×
from 136Xe, T0νββ(136Xe) > 1.07 1026yr [36], which, using the matrix elements from [33], translates to
1/2 ×
(cid:15) <4 10−9 at 90% C.L.
3
×
On the other hand, upcoming experiments are expected to be sensitive to lifetimes of order 1027–1028 yrs [37],
i.e. a reduction factor on the coupling of about one order of magnitude. Thus, for 0νββ decay mediated by
heavy particles to be observable in the next round of experiments we should have (cid:15) (cid:38)4 10−10. Therefore in
3
×
order to escape the current experimental bounds but at the same time to entertain the possibility of observing
0νββ decay in the near future, we require (cid:15) to be within the following range:
3
4 10−10 <(cid:15) <4 10−9. (23)
3
× ×
With f , λ 1 and µ m m 1TeV and I 1 we obtain, from Eq. (22), (cid:15) 10−8. But it
ee 6 κ A κ++ β 3
≈ ≈ ≈ ≈ ∼ ∼
is easy to arrange a suppression through a better choice of input parameters so that (cid:15) falls within the range
3
given in Eq. (23). Later in this article we shall provide one such benchmark as an example.
4 Estimation of the neutrino masses
From Eqs. (2) and (3) it is obvious that simultaneous nonzero values for Y , f , µ and λ will prevent us
e ab κ 6
fromassigningconsistentleptonnumberstoallthescalarandleptonfields. Therefore,leptonnumberisbroken
explicitly and Majorana neutrino masses will be unavoidable. The sample diagram of Fig. 2, in the mass
5
Φ σ Φ
h i h i
χ χ
− −
Φ Φ
− −
νL κ−− νL
e e
R R
Figure 2: Samplethreeloopdiagram, inthemassinsertionapproach, contributingtotheneutrinomasses.
insertion approach, clearly depicts the involvement of all these couplings in a multiplicative manner. Thus, we
can parametrize the neutrino mass matrix as follows:
8µ λ2
M = κ 6 I m f m , (24)
ab (4π)6m2 ν a ab b
κ++
where m denotes the mass of the charged lepton, (cid:96) , and I represents the loop function expected to be of
a a ν
(1). Detailed expression of I in terms of the scalar masses has been presented in Appendix B. Eq. (24) has
ν
O
a very particular and predictive structure, specific for this class of models, which can be constrasted with the
observed spectrum of neutrino masses and mixings (see for instance Refs. [26,27,38]).
As before, taking f , λ 1 and µ m 1 TeV and I 1 we obtain the following values for the
ττ 6 κ κ++ ν
≈ ≈ ≈ ∼
different elements
M 10−7eV, M 10−4eV, M 10−3eV, M 10−2eV, M 10−1eV, M 10eV. (25)
ee eµ eτ µµ µτ ττ
∼ ∼ ∼ ∼ ∼ ∼
But of course, some of the f s can be much smaller than 1. However, not all of the elements of the f matrix
ab
are arbitrary as some of them will be constrained from LFV processes. We will discuss these constraints in
Sec. 5. But for now we wish to emphasize that the product f∗f will receive strong bounds from µ 3e
| ee eµ| →
as the latter can proceed at the tree-level mediated by κ++. Then, one should naturally expect the following
hierarchy among the mass matrix elements:
M ,M M ,M ,M ,M , (26)
ee eµ eτ µµ µτ ττ
(cid:28)
which, obviously, can only accommodate a normal hierarchy among the neutrino masses. In Ref. [27] it has
been shown that the above hierarchy with
3M M M M 0.02 eV (27)
eτ µµ µτ ττ
∼ ∼ ∼ ∼
cansuccessfullyreproducetheobservedmassesandmixingsintheneutrinosectorwithapredictionofsin2θ >
13
0.008. Eq. (27) will imply the following hierarchy among the Yukawa elements:
m m2 m
3f τf >f τf >f τf >f . (28)
eτ ∼ m ττ µµ ∼ m2 ττ µτ ∼ m ττ ττ
e µ µ
We shall also assume f f in such a way that f∗f is still sufficiently small to keep µ 3e decay under
ee (cid:29) eµ ee eµ →
control but at the same time allowing for the possibility of large 0νββ decay.
From Eqs. (22) and (24) we see that the dimensionless factor,
µ λ2 2sin2αcos2α(m2 m2)2 µ
γ = κ 6 = H − S κ , (29)
m v4 m
κ++ κ++
is common to both. In terms of γ, the explicit expression for M in Eq. (27) reads:
ττ
8 m2f
M = γI τ ττ 0.02 eV. (30)
ττ (4π)6 ν m ≈
κ++
6
As we will see in Sec. 5, the ratio f /m is bounded from LFV processes as f /m (cid:46)1.4 10−4 TeV−1.
ττ κ++ ττ κ++
×
Plugging this into Eq. (30) we obtain the following bound for γ:
22
γ (cid:38) . (31)
I
ν
5 Constraints from LFV processes
Experimental Data (90% CL) Bounds (90% CL) Bounds assuming Eq. (28)
BR(µ− →e+e−e−)<1.0×10−12 |feµfe∗e|<2.3×10−5 (cid:0)mTκe+V+(cid:1)2
BR(τ− →e+e−e−)<2.7×10−8 |feτfe∗e|<0.009 (cid:0)mTκe+V+(cid:1)2 |fe∗efττ|(cid:46)7.8×10−6 (cid:0)mTκe+V+(cid:1)2
BR(τ− →e+e−µ−)<1.8×10−8 |feτfe∗µ|<0.005 (cid:0)mTκe+V+(cid:1)2 |fe∗µfττ|(cid:46)4.3×10−6 (cid:0)mTκe+V+(cid:1)2
BR(τ− →e+µ−µ−)<1.7×10−8 |feτfµ∗µ|<0.007 (cid:0)mTκe+V+(cid:1)2 |fττ|(cid:46)1.4×10−4 (cid:0)mTκe+V+(cid:1)
BR(µ→eγ)<5.7×10−13 |fe∗ef<eµ1+f1e∗0µ−f7µµ(m+κ+f+e∗τ)f4µτ|2 |fττ|(cid:46)1.2×10−4 (cid:0)mTκe+V+(cid:1)
× TeV
Table 2: Relevant constraints for our model from LFV decays [39,40]. Limits on the Yukawa couplings
of the doubly charged singlet scalars have been taken from Ref. [41]. The constraints in the third column
are obtained from those in the second column assuming Eq. (28) holds. The bound in the third column
corresponding to µ eγ has an additional assumption, f 0.
eµ
→ ≈
ConstraintsfromLFVprocessescomemainlyfromdecaysofthetype(cid:96)∓ (cid:96)±(cid:96)∓(cid:96)∓ and(cid:96)∓ (cid:96)∓γ. Inourcase
(cid:96)∓ (cid:96)±(cid:96)∓(cid:96)∓ will be more important because these decays can occur aat t→he tbrece-ldevel thraou→gh tbhe exchange of
a → b c d
the doubly charged scalar singlet, κ±±. These processes along with the kinds of constraints they imply have
been reviewed inRef.[41] in the context of theZee-Babumodel (see also Refs. [30,42]). The experimental data
has not changed much since then. In the first two columns of Table 2 we have summarized the experimental
data and the corresponding constraints on the Yukawa couplings. In the third column of Table 2 we recast the
constraints of the second column assuming the validity of Eq. (28). This allows us to express the constraints
in more specific forms. For example, using m f m f and m2f m2f , the constraint from τ eµµ
e eτ ∼ τ ττ µ µµ ∼ τ ττ →
leads to a direct bound on f as follows:
ττ
(cid:16)m (cid:17)
f (cid:46)1.4 10−4 κ++ . (32)
| ττ| × TeV
It is also worth mentioning that, using Eq. (28), the limit from τ 3e translates into
→
(cid:16)m (cid:17)2
f∗f (cid:46)7.8 10−6 κ++ . (33)
| ee ττ| × TeV
As mentioned earlier, we want to have f relatively large to have appreciable 0νββ decay rate in the future
ee
experiments. Then we will need f to be vanishingly small to keep the constraints from µ 3e under control.
eµ
→
Note that, for f (1) and sub TeV κ++, Eq. (33) will imply a stronger bound on f than Eq. (32).
ee ττ
∼O
6 Dark Matter
Our model has a Z symmetry which remains unbroken after the SSB. Consequently, the particle spectrum can
2
be divided into Z -even and odd sectors as shown in Table 1. Among the Z odd neutral scalars, S, being the
2 2
7
lightest,isapromisingcandidateforDM.NoticethatS isandadmixtureoftherealsingletandthetriplet,and
therefore, it will feel both, Higgs and gauge interactions4. In spite of that, one can parametrize its couplings
with the SM-like Higgs boson as follows:
(cid:18) (cid:19)
L ⊃−12λSS2(cid:12)(cid:12)Φ0(cid:12)(cid:12)2 ⊃−21λSS2 vh+ 12h2 , (34)
1(cid:104) (cid:105)
with, λ = 2λ cos2α 2√2λ sinαcosα+(λ +λ(cid:48) )sin2α . (35)
S 2 Φσ − 6 Φχ Φχ
Figure 3: Regions corresponding to the observed relic abundance [45] in the m -λ plane for different
S S
values of sinα. We have chosen m = m = m = 800 GeV as a benchmark for this plot. Current
H χ++ κ++
[46,47] and future [48] bounds from direct detection experiments are also marked appropriately.
InFig.3wehavedisplayedregionsinthem -λ plane,whichcanreproducetheobservedDMrelicdensity[45].
S S
For this plot, we have assumed m = m = m = 800 GeV and used the MicrOMEGAs package [49] to
H χ++ κ++
compute the DM abundance. Note that, the region labeled as sinα = 0 corresponds to the pure Higgs portal
scenario. Barring the small window near the Higgs-pole (m m /2, not shown explicitly in the plot), in
S h
this case, we need m (cid:38) 350 GeV [50,51] to evade the direct≈search bound. It is worth mentioning that in
S
the case of pure Higgs portal, for our choice of benchmark, the DM annihilates through ff¯, WW, ZZ and hh
mainly. All these annihilation channels except hh can only proceed through s-channel h exchange. But as sinα
is turned on, we allow for a direct SSVV (V = W,Z) with strength proportional to g2sin2α. For our choice
of positive values for λ , the new contact diagram will interfere constructively with the h mediated s-channel
S
diagram.5 This will enhance the annihilation rate for SS VV once the corresponding threshold is reached.
→
Therefore, we would require lower values of λ , compared to the pure Higgs portal case, to reproduce the relic
S
abundance. ThesefeatureshavebeendepictedinFig.3wherewecanseethatasmallvalueofsinαissufficient
to accommodate DM with mass as low as 200 GeV, which can either be discovered or ruled out in the next run
of direct detection experiments.
4ForrecentstudiesofaDMcandidatewhichisanadmixtureofascalarsingletandaY=0tripletseeforinstance[43,44].
5 Anonzerovalueofsinαwillalsoinducet-channeldiagramsforSS→VV,hhmediatedbyχ±,AorH. Buttheseamplitudes
willbesuppressedaslongasmχ+,mA,mH (cid:29)mS. Alsonotethat,inthislimit,thegaugecouplingsofS donotcontributetothe
directdetectioncrosssection[52,53].
8
7 Results and conclusions
m (GeV) m (GeV) sinα m (GeV) m (GeV) µ (TeV) f f f
χ++ κ++ H S κ ee ττ eµ
| | | | | |
800 800 0.08 800 200 20 0.01 10−4 0
Table 3: Benchmark values for the input parameters.
m (GeV) m (GeV) I I (cid:15) f f f
χ+ A β ν 3 eτ µµ µτ
| | | | | |
799 798 0.165 0.84 3.5 10−9 0.12 0.03 1.7 10−3
× ×
Table 4: Other relevant parameters computed using the inputs from Table 3.
Since κ±± couples directly to the charged leptons, it will be strongly constrained from the same sign dilepton
searchesattheLHC.Dependingonthepreferreddecaychannelofκ±±, theboundcanbeasstrongasm (cid:38)
κ++
500 GeV [54,55]. On the other hand, to keep the T-parameter under control, for small sinα, we will need
m m (cid:46)100 GeV (see Eq. (16)). All these considerations together justifies our choice of benchmark for
H χ++
| − |
Fig. 3. Now, to satisfy Eq. (31) we need to have a large splitting between m and m . Keeping these things in
H S
mind, we have chosen Table 3 as a benchmark for the input parameters. The output parameters that follows
fromTable3havebeendisplayedinTable4. FromthenumbersofTables3and4onecaneasilycheckthatthe
constraints of Eqs. (23) and (31) and all the bounds in Table 2 are satisfied. Moreover, using Eq. (28) suitable
values for f , f and f can be found so that the hierarchy of Eq. (27) is satisfied. Note that Eq. (28) will
eτ µµ µτ
alsoentailanapproximatecorrelationamongdifferentLFVdecayrates,whichcanbetakenasatypicalfeature
of our model.
Tosummarize,wehavedemonstratedthatourmodeliscompatiblewiththeneutrinodataandcaneasilysatisfy
alltheconstraintsarisingfromLFVdecays. Ascanbeseenfromthevalueof(cid:15) inTable4,ourmodelopensup
3
the interesting possibility of detecting 0νββ decay in the next generation of experiments. We have also found a
DM candidate which can reproduce the observed relic abundance yet can survive the current constraints from
the direct detection experiments. Furthermore, our model provides the prospect of detecting new scalars with
masses below (TeV), in the future collider experiments (for LHC studies on lepton number violating singly
O
and doubly charged scalars see for instance [56,57]). Among these new particles, χ± and χ±± being Z -odd,
2
cannot decay directly into the SM particles. A search strategy for these kinds of exotic charged scalars can be
interesting for the collider studies.
Acknowledgements
A.S. would like to acknowledge J.M. No for discussions. All Feynman diagrams have been drawn using Jaxo-
Draw [58,59]. This work has been partially supported by the Spanish MINECO under grants FPA2011-23897,
FPA2014-54459-P, by the “Centro de Excelencia Severo Ochoa” Programme under grant SEV-2014-0398 and
by the “Generalitat Valenciana” grant GVPROMETEOII2014-087.
Appendix A Computation of the loop induced κWW vertex
Herewecomputetheeffectiveκ−−W+Wµ+ vertexatoneloopforvanishingexternalmomenta. Ourassumption
µ
is justified in view of the fact that the momentum transfers to κ and W-bosons in Fig. 1 are much smaller than
9
W− S,A,H W− S,A,H W−
χ
−
κ−− κ−− κ−−
S,A,H χ
−
χ− W− χ−− W− χ−− W−
Figure 4: One loop diagrams contributing to the κWW vertex in the unitary gauge.
the corresponding masses. We write the effective vertex as
L = C κ−−W+Wµ++h.c., (A.1)
κWW κWW µ
which, after spontaneous symmetry breaking, emerges from the following gauge invariant operator:
(cid:16) (cid:17)(cid:16) (cid:17)
L =C κ++ Φ†DµΦ˜ Φ†D Φ˜ +h.c. (A.2)
κeff κeff µ
After integrating out κ++, Eq. (A.2) leads to the following LFV gauge invariant operator [26,27]:
(cid:16) (cid:17)(cid:16) (cid:17)
L =C (e f∗ ec) Φ†DµΦ˜ Φ†D Φ˜ . (A.3)
eeWW eeWW R ee R µ
We depict in Fig. 4 the three diagrams that contribute to the vertex. Each of these diagrams seem to diverge
logaritmically. But one should keep in mind that the neutral scalar exchange must violate lepton number
conservation. Thus a large cancellation among the contributions from the three neutral scalars, A, H and S,
is expected. After adding all the contributions we obtain an effective neutral scalar propagator of the following
form (for Minkowsky momenta)
1 sin2αcos2α(m2 m2)2 λ2 Φ 4
H − S = 6(cid:104) (cid:105) , (A.4)
2(p2 m2 )(p2 m2)(p2 m2) (p2 m2 )(p2 m2)(p2 m2)
− H − S − A − H − S − A
where, Φ = v/√2. Evidently, after adding contributions from A, H and S, every diagram in Fig. 4 becomes
(cid:104) (cid:105)
finite individually. Now we can write the expression of C (defined in Eq. (A.1)) as follows:
κWW
1
C = µ g2λ2 Φ 4 I , (A.5)
κWW κ 6(cid:104) (cid:105) 16π2m4 β
A
where I is a function of the masses of the particles running in the loop and contains three contributions
β
corresponding to the three diagrams in Fig. 4. Thus, we express I as follows:
β
I = I1+I2+I3, with, (A.6)
β β β β
(cid:90) ∞ q2
I1 = m4 dqq3 , (A.7)
β A (q2+m2 )2(q2+m2)(q2+m2 )(q2+m2)
0 χ+ A H S
(cid:90) ∞ 1
I2 = 2m4 dqq3 , (A.8)
β − A (q2+m2 )(q2+m2)(q2+m2 )(q2+m2)
0 χ++ A H S
(cid:90) ∞ q2
I3 = 2m4 dqq3 , (A.9)
β A (q2+m2 )(q2+m2 )(q2+m2)(q2+m2 )(q2+m2)
0 χ++ χ+ A H S
where we have passed to Euclidean momenta and integrated over the angular variables. Adding the three
contributions we simplify the expression for I as follows:
β
(cid:90) ∞ q4+q2(m2 2m2 ) 2m4
I =m4 dqq3 χ++ − χ+ − χ+ . (A.10)
β A (q2+m2 )(q2+m2 )2(q2+m2)(q2+m2 )(q2+m2)
0 χ++ χ+ A H S
We have also checked that we obtain the same result by using the equivalence theorem where the external
W-bosons are replaced by the corresponding Goldstone bosons.
10