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IL NUOVO CIMENTO Vol. ?, N. ? ?
Right-Handed Neutrinos: DM and LFV vs. Collider
M. Chekkal(1), A. Ahriche(2), A.B. Hammou(1) and S. Nasri(3)
(1) Department of Physics, University of Sciences and Technology of Oran, BP 1505, Oran, ElM’Naouer, Algeria.
(2) Department of Physics, University of Jijel, PB 98 Ouled Aissa, DZ-18000 Jijel, Algeria.(3) Department of physics, United Arab Emirates University, Al-Ain, UAE.
Summary. — In a class of neutrino mass models with a lepton flavor violation(LFV) Yukawa interaction term that involves a heavy right handed neutrino, acharged scalar and a charged lepton, we investigate at the ILC@500 GeV the possi-bility of observing news physics. These models can address neutrino mass and darkmatter without being in conflict with different LFV constraints. By imposing DMrelic density and LFV constraints, we recast the analysis done by L3 collaborationat LEP-II of monophoton searches on our space parameter and look for new physicsin such channels like monophoton and SS(γ), where we give different cuts and showthe predicted distributions. We show also that using polarized beams could improvethe statistical significance.
1. – Introduction
Neutrino oscillations have been put in evidence by different experiences and observa-tions caused by nonzero neutrino masses and neutrino mixing [1]. However, the StandardModel (SM) does not explain the intrinsic properties of neutrinos such as their origin,nature and the smallness of their masses. The seesaw mechanism [2] is the most popularmethod to explain the tiny mass of SM neutrinos but poses scale problems and preventsthe direct detection of the right-handed (RH) neutrinos introduced by this mechanismbecause of its large mass compared to the electroweak scale.The radiative neutrino mass models [3, 4, 5, 6, 7] is another way to generate a small massto light neutrinos at loop level and to circumvent the scale problem. The violation ofthe leptonic number is permitted by the fact that the neutrinos are Majorana particleswhere the lightest RH neutrino is identified as being the dark matter and have large phe-nomenological implications. We can take as an example the model in [8] where authorsshow that the scale of new physics can be in the sub-TeV for the 3-loops neutrino massgeneration model [6] which makes it testable at collider experiments [9]. In this work, westudy the possibility of detecting the manifestations of the new physics resulting fromthis class of radiative neutrino mass models.
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2 M. CHEKKAL, A. AHRICHE, A.B. HAMMOU and S. NASRI
2. – LFV and DM Constraints Class of Models with RH Neutrinos
A class of radiative neutrino mass models are considered here, wich are extending theSM with three right-handed neutrinos Ni (i = 1, 2, 3) and a SU(2)L-singlet charged scalarS±. The models contain the following Yukawa term in in the Lagrangian [5, 6, 7, 10]
(1) LN ⊃ −1
2mNiN
ci PRNi + giαS
+Ni`αR+ h.c.,
where `αRis the right-handed charged lepton and giα are Yukawa couplings. The stability
of the lightest RH neutrino, wich is supposed play the DM role, is assured by imposingthe global Z2 symmetry(1). `α → `βγ and `α → 3`β are LFV processes produced by thistype of interaction.
The contribution of the interactions (1) to the `α → `βγ branching ratio is givenby [12]
(2) B(N)(`α → `βγ) =3(4π)3α
4G2F
|AD|2 × B(`α → `βνανβ),
where α is the electromagnetic fine structure constant, GF is the Fermi constant and ADis the dipole contribution given by
(3) AD =
3∑i=1
g∗iβgiα
2(4π)21
m2S
F (xi) ,
with xi = m2Ni/m2
S and F (x) is a loop function.We scanned over all the free parameters of our model to determine the the phenomeno-
logical implications for the dark matter and the searches of new Physics at colliders. Toget a feeling of the different contributions from the RH neutrino couplings, we define theratio (fine-tuning parameter)
(4) R =|∑3i=1 g
∗iµgieF (xi) |2
Max[| g∗iµgieF (xi) |2],
which represents the way the cancellation between different combinations g∗iβgjα oc-curs in order to suppress the LFV branching ratios even for large g-couplings. Forinstance, the parameter R could be signicantly smaller than unity due to possible cancel-lation between different RH neutrinos contributions, and this may allow the g-couplingsto be relatively large.
In Fig. 1, for different values of fine-tuning parameter R ≈ 1, 10−2, 10−4, thebranching ratios for the processes `α → `βγ and `α → 3`β versus the charged scalarmass are presented. The experimental bounds of all branching fractions are the firstconstraint to be obeyed by the free parameters of the model, and some hundreds ofconfigurations of free parameters are generated in this way.
(1) The global Z2 symmetry is accidental for higher representation (setplet) of RH neutri-nos [11].
RIGHT-HANDED NEUTRINOS: DM AND LFV V S. COLLIDER 3
10-17
10-16
10-15
10-14
10-13
10-12
10-11
102
103
B(µ
->
eγ)
mS (GeV)
R~ 1R~ 10
-2
R~ 10-4
10-14
10-13
10-12
10-11
10-10
10-9
10-8
10-7
10-6
102
103
B(τ
->
µγ)
mS (GeV)
R~ 1R~ 10
-2
R~ 10-4
10-12
10-11
10-10
10-9
10-8
10-7
10-6
102
103
B(τ
->
e γ
)
mS (GeV)
R~ 1R~ 10
-2
R~ 10-4
10-17
10-16
10-15
10-14
10-13
10-12
10-11
102
103
B(µ
->
3 e
)
mS (GeV)
R~ 1R~ 10
-2
R~ 10-4
10-18
10-16
10-14
10-12
10-10
10-8
102
103
B(τ
->
3 µ
)
mS (GeV)
R~ 1R~ 10
-2
R~ 10-4
10-13
10-12
10-11
10-10
10-9
10-8
10-7
10-6
102
103
B(τ
->
3 e
)
mS (GeV)
R~ 1R~ 10
-2
R~ 10-4
Fig. 1. – The branching ratios (top) B(µ → eγ), B(τ → µγ) and B(τ → eγ); and (bottom)B(µ→ 3e), B(τ → 3µ) and B(τ → 3e) versus mS . The horizontal dashed lines show the currentexperimental upper bounds for each radiative decay.
As mentioned earlier, the dark matter condidate could be the lightest RH neutrinosN1 which is supposed stable. We can safely keep only the contribution of N1 densityand neglect that of N2 and N3 in hierarchical RH neutrino mass spectrum case. Theannihilation process N1N1 → `α`β via t-channel exchange of S± impoverished the densityof N1. When the temperature of the universe drops below the freeze-out temperature,and using the equation Boltzmann equation, we can approximate the relic density afterthe decoupling of N1 from the thermal bath [13]
(5) ΩN1h2 ' 2xf × 1.1× 109 GeV−1√
g∗Mpl 〈σN1N1vr〉' 17.56∑
α,β |g1αg∗1β |2( mN1
50 GeV
)2 (1 +m2S/m
2N1
)41 +m4
S/m4N1
,
in Fig. 2, we present a contour plot mN1versus mS , where in palette we have the
coupling combination∑αβ
∣∣∣g1αg∗1β∣∣∣2, which appears in the expression of the relic density,
within the conditions mN1< mS and mS > 100 GeV being imposed. It is difficult to
maintain all LFV ratios within the current experimental bounds for values of couplingcombination larger than 10, and required an extreme fine-tunning. So, once the relicdensity are imposed and mN1
and mS are defined the∑αβ |g1αg1β |
2impose another
condition in addition to LFV constraint and the most viable range of the masses isextracted as mN1
< 200 GeV and mS < 300 GeV.
3. – Constraints from LEP-II
An additional constraint on the free parameters is imposed by the no evidence formassive neutral particle realized by the L3 detector at LEP-II [14], which has conductedan analysis on single and multi photon events with missing for center of mass energiesbetween 189 and 209 GeV. Indeed, benchmark points that respect the different DMand LFV constraints together must also give non-revelent significance under the sameconditions as those of LEP.
In the next sections, we will carried out the electron-positron (electron-electron) col-lision on the ILC, so the decay length of the unstable particles N2 and N3 must then
4 M. CHEKKAL, A. AHRICHE, A.B. HAMMOU and S. NASRI
be measured to determine whether they are disintegrating inside or outside the detec-tors. From Fig. 3, It can be seen that N3 does not contribute to the missing energy,because being disintegrated mainly inside the detectors, whereas a substantial amountof N2 events escape from the detector. In all our analysis the benchmarks points areverified in order to precisely identify the missing energy.
0
100
200
300
400
500
100 200 300 400 500 600 700 800 900 1000
mN
1 [G
eV
]
mS [GeV]
10-1
100
101
102
103
104
Fig. 2. – Dark Matter mass versus the charged scalar mass, The palette represents the quantity∑αβ
∣∣g1αg∗1β∣∣2. The dashed curves (from left to right) represent the values∑αβ
∣∣g1αg∗1β∣∣2 =1, 10, 100, respectively.
10-10
10-5
100
105
1010
1015
20 40 60 80 100 120 140
De
ca
y le
ng
th [
m]
mN2 [GeV]
100
120
140
160
180
200
220
240
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
102
20 40 60 80 100 120 140 160
De
ca
y le
ng
th [
m]
mN3 [GeV]
100
120
140
160
180
200
220
240
Fig. 3. – The decay length of the RH neutrinos N2 (left) and N3 (right) as a function of mN2
and mN3 , respectively. The palette represents the charged scalar mass mS [GeV].
We consider the highest integrated luminosities 176 pb−1 and 130.2 pb−1 at the centerof mass energies
√s = 188.6 GeV and
√s = 207 GeV, respectively. Same kinematical
cuts used by L3 collaboration for a high energy single photon are applied [14]: |cos θγ | <0.97, pγt > 0.02
√s and Eγ > 1 GeV. We compute the cross sections of the signal
e−e+ → γ + Emiss and the background e−e+ → νiνjγ using the LanHEP/CalcHEPpackages [15, 16], for thousands of aforementioned benchmark points.
The results are shown in Fig. 4, where in palette one can reads ∆, a quantity at whichthe cross-section is sensitive. An exclusion bound on a combination of these parameteris drived according to LEP analysis and the significance S must be smaller than three,thereby we extract the following constraint,
(6) ∆ =∑i,k
|gieg∗ke|2
[150 GeV
mS
] [50 GeV√mNi
mNk
]< 1.95.
RIGHT-HANDED NEUTRINOS: DM AND LFV V S. COLLIDER 5
10-6
10-4
10-2
100
102
104
10 20 30 40 50 60 70 80 90
σ (
e+e
- ->
γ +
Em
iss )
mN1 [GeV]
√s = 188.6 GeV , L = 176 pb-1
10-2
10-1
100
101
10-6
10-4
10-2
100
102
104
10 20 30 40 50 60 70 80 90
σ (
e+e
- ->
γ +
Em
iss )
mN1 [GeV]
√s = 207.2 GeV , L = 130.2 pb-1
10-2
10-1
100
101
Fig. 4. – The cross section for the randomly chosen benchmark points for the process e−e+ →γ + Emiss at LEP as a function of mN1 for the CM energies
√s = 188.6 GeV (right) and√
s = 207.2 GeV (left). The palette represents the combination ∆, and the black dashed linescorrespond to S = 2, 3, respectively. The red dashed line corresponds to the background.
4. – Possible Signatures at Lepton Colliders
In this work, we are interested in the possibility of probing/detecting the traces of thenew physics mediated by the charged scalar and giving dark mater in the final state, atlepton collider specially the International Linear Collider (ILC) [17] which covers centerof mass (CM) energies from 250 to 500 GeV. We study the most interesting signatures
10-10
10-8
10-6
10-4
10-2
100
102
0 20 40 60 80 100
σ (
e+e
- ->
γ +
Em
iss )
[pb]
mN1 [GeV]
R1 R2 R3
10-4
10-3
10-2
10-1
100
101
102
103
104
80 100 120 140 160 180 200 220 240 260
σ(e
+e
- ->
S+S
- ) [p
b]
mS [GeV]
R ~ 1 Drell-Yan R ~ 10-2
R ~ 10-4
10-6
10-5
10-4
10-3
10-2
10-1
100
101
102
80 100 120 140 160 180 200 220 240 260
σ(e
+e
- ->
S+S
- γ)
[pb]
mS [GeV]
R ~ 10-2
R ~ 10-4 R ~ 1 Drell-Yan
10-6
10-5
10-4
10-3
10-2
10-1
100
101
102
103
0 20 40 60 80 100
Sig
nific
ance (
e+e
- ->
γ +
Em
iss )
mN1 [GeV]
R1 R2 R3
10-2
10-1
100
101
102
103
80 100 120 140 160 180 200 220 240 260
Sig
nific
ance (
e+e
- ->
S+S
- )
mS [GeV]
R ~ 1 R ~ 10-2
R ~ 10-4
10-5
10-4
10-3
10-2
10-1
100
101
102
80 100 120 140 160 180 200 220 240 260
Sig
nific
ance (
e+e
- ->
S+S
- γ)
mS [GeV]
R ~ 1 R ~ 10-2
R ~ 10-4
10-5
10-4
10-3
10-2
10-1
100
101
102
103
104
80 100 120 140 160 180 200 220 240 260
σ(e
- e- -
> S
- S- )
[pb]
mS [GeV]
R ~ 1 R ~ 10-2
R ~ 10-4
100
101
102
103
80 100 120 140 160 180 200 220 240 260
Sig
nific
ance (
e-e
- ->
S- S
- )
mS [GeV]
R ~ 1 R ~ 10-2
R ~ 10-4
Fig. 5. – The cross section values (top) and the corresponding significance values (middle) forproduction via electron-positron collision and at bottom for production via electron-electroncollision at luminosity 100 pb−1 in function of mS . The red lines represent the background valueand the dashed one represents the Drell-Yann contribution in cross section, and the dashed linesrepresent S = 3, 5 in significance, respectively.
6 M. CHEKKAL, A. AHRICHE, A.B. HAMMOU and S. NASRI
Table I. – Three benchmark points selected from the parameters space of the model.
Point B1 (R1) B2 (R2) B3 (R3)
g1e (7.506 + i0.014)× 10−1 (1.8284 + i0.103) (−0.103 + i0.201)
g2e (−0.26819− i1.5758)× 10−4 (1.543 + i3.004)× 10−4 (0.654− i2.616)× 10−2
g3e (−1.360− i0.707) (0.313− i0.549) (−0.869− i0.878)
mS(GeV) 196.75 242.81 104.47
mN1(GeV) 25.788 43.764 38.306
mN2(GeV) 28.885 58.182 56.481
mN3(GeV) 36.274 67.511 72.440
which occure via the interactions in (1), and give the following processes:
e−e+ → γ + Emiss,
e−e+ → S+S− → `+α `−β + Emiss,
e−e− → S−S− → `−α `−β + Emiss
e−e+ → γ + S+S− → γ + `+α `−β + Emiss,(7)
Three processes for electron-positron collision are analyzed; photon(s) with a pair ofDM in the final state where the background contributing to the signal give left-handedneutrinos with photon, and a pair production of charged scalars S+S− with or withouta photon in the final state, where each charged scalar decays into a RH neutrino and acharged lepton. The corresponding background comes from the process e+e− →W+W−
where W decays into a light neutrino and a charged lepton. Another potential signaturecome from electon-electron collison and give same sign pair of charged scalars. A firstqualitative analysis is carried out on the four processes in (7), on three sets of bench-mark points according to different values of the ratio R for a center of mass energy of√s = 500 GeV and with a luminosity L = 100 pb−1. The cross section values and the
corresponding significance are shown in Fig. 5 versus the charged scalar mass withoutapplying any cut(2). One remarks that the cross section values of the processes (7) inFig. 5 varies over seven orders of magnitudes as its sensitively depends on our choice ofthe parameters space. The production cross section via electron-electron collision is largecompared to the background, so even for low luminosity the significance is huge. Hence,same sign pair of charged scalars process is a clean and RH neutrinos can be directlyprobed at ILC. Detectability of production via electron-positron collision is developed inmore detail in the next chapter.
(2) Except the cut Eγ > 8 GeV and |cos θγ | < 0.998 on channels with photon in finale state.
RIGHT-HANDED NEUTRINOS: DM AND LFV V S. COLLIDER 7
0.1
1
10
100
100 150 200 250 300 350 400 450 500
Sig
nific
an
ce
( e
- e+ -
> γ
+ E
mis
s )
mS [GeV]
B1B2B3
1
10
100
100 120 140 160 180 200 220 240 260 280
Sig
nific
an
ce
( e
- e+ -
> S
+S
- )
mS [GeV]
B1B2B3
0.1
1
10
100
100 120 140 160 180 200 220 240 260 280
Sig
nific
an
ce
( e
- e+ -
> S
+S
- γ )
mS [GeV]
B1B2B3
Fig. 6. – The signal significance for the processes e−e+ → γ + Emiss (left), e−e+ → S−S+
(middel) and e−e+ → S−S+ + γ (right) as a function of mS for the values of giα given Table. Iat integrated luminosity of 1/10/10 (solid), 5/50/100 (dashed) and 10/100/500 (dash-dotted)fb−1 respectively. The horizontal dashed lines correspond to a 3 and 5 sigma significance. Forthe values mS > 250 GeV, the charged scalars is off-shell.
5. – Benchmark Analysis
Let’s now consider three benchmarks points, one of each ratio R1 ≈ 1, R2 ≈ 10−2 andR3 ≈ 10−4, with nearby heavy neutrinos masses relatively. As can be seen on the table.I,our freedom of the model parameters space are substantially limited by the choice of theratios Ri.The distributions for different kinematic variables are generated for signal andbackground Using CalcHEP [16] for the processes e−e+ → γ + Emiss, e
−e+ → S−S+,and e−e+ → S−S+ + γ at 500 GeV. We extract the optimal kinematical cuts for eachprocess and this can be achieved as the following,
Final state γ + Emiss : 8 GeV < Eγ < 300 GeV, |cos θγ | < 0.998and Emiss > 300 GeV.
Final state S+S− : M`+,`− < 300 GeV, 150 GeV < Emiss < 420 GeV,30 GeV < E` < 180 GeV and p`t < 170 GeV.
Final state S+S−γ : M`+,`− < 300 GeV, 150 GeV < Emiss < 400 GeV,30 GeV < E` < 170 GeV, p`t < 170 GeV, |cos (θγ) | < 0.5,
8 GeV < Eγ < 120 GeV and pγt < 110 GeV.
By varying the charged scalar mass, the significance for these processes for the threeconsidered benchmark points are shown in Fig. 6 at integrated luminosity that brings uscloser to 5 sigma significance for each channels. For the monophoton process, the signal
10-1
100
101
102
103
0.1 1 10 100 1000
Sig
nific
ance
L [fb-1
]
P(e-,e
+)=[0,0]
e+e
- -> γ + Emiss
e+e
- -> S
+S
-
e+e
- -> γ + S
+S
-
10-1
100
101
102
103
0.1 1 10 100 1000
Sig
nific
ance
L [fb-1
]
P(e-,e
+)=[+0.8,-0.3]
e+e
- -> γ + Emiss
e+e
- -> S
+S
-
e+e
- -> γ + S
+S
-
Fig. 7. – The signal significance as function of luminosity for the three different signaturesstudied, at left without polarization and at right with polarization of P (e−, e+)=[+0.8,−0.3].The two horizontal dashed lines correspond to to a 3 and 5 sigma significance, respectively.
8 M. CHEKKAL, A. AHRICHE, A.B. HAMMOU and S. NASRI
significance becomes detectable at the ILC for an integrated luminosity of a few hundredfb−1. A luminosity of a few ten fb−1 is necessary to probe the production of a paircharged scalars with a photon while the S−S+ channel could be visible easily at verylow luminosity, around 0.5 fb−1. The charged scalar mass must be lighter than about220 GeV in all these channels
There is an additional feature allowing the improvement of detection and which isavailable on the ILC, the possibility to have highly polarized electron/positron beams. alongitudinal polarization of 80% for the electron beam and 30% for the positron beamare planned by the ILC. We re-analyze the processes discussed earlier with all the pos-sible polarizations combinations in order to improve the signal-background ratio and wefound that for polarized beams as P (e−, e+) = [+0.8,−0.3] while applying the samecuts used previously, the number of background events gets reduced by 86% and thesignal increased by 130%. In Fig. 7, we present the significance for P (e−, e+)=[0, 0]and P (e−, e+)=[+0.8,−0.3], for the benchmark point B3 as a function of luminosity.one can note that the signal over background gets improved and the required integratedluminosity is dictated by a factor about ten for each processes studied.
6. – Conclusion
This paper investigates the behavior of some type of interaction present in a classof models that extends the standard model by majorana right-handed neutrinos andsclaire charged, to explain violations of leptonic flavor on one side, and to give a seriouscondidat to the dark matter of another. After imposing several constraints on the freeparameters of models and carrying out an in-depth analysis on the detectability of themajor processes resulting from the electron-positron collision in the conditions of thefuture lepton collider : ILC, we show that this type of interaction is likely to be probed,for different luminosity according to the final state, but which are all largely within thescope of the ILC capacity. It is also shown that the polarization of the electon / positronbeams present in this collider can lead to a positive result very quickly.
* * *
MC wants to thank the organizers for the finanicial support. AA is supported bythe Algerian Ministry of Higher Education and Scientific Research under the CNEPRUProject No B00L02UN180120140040.
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