A three-loop radiative neutrino mass model with dark matter

0.44 MB·English

by  Li-Gang Jin

A three-loop radiative neutrino mass model with dark matter
Li-Gang Jin,1,∗ Rui Tang,1 and Fei Zhang1
1Department of Physics and Institute of Theoretical Physics,
Nanjing Normal University, Nanjing, Jiangsu 210023, Peoples Republic of China
Wepresentamodelthatgeneratessmallneutrinomassesatthree-looplevelduetotheexistence
of Majorana fermionic dark matter, which is stabilized by a Z symmetry. The model predicts
2
thatthelightestneutrinoismassless. Weshowaprototypicalparameterchoiceallowedbyrelevant
experimental data, which favors the case of normal neutrino mass spectrum and the dark matter
withm∼50−135GeVandasizableYukawacoupling. Itmeansthatnewparticlescanbesearched
for in future e+e− collisions.
Keywords: neutrinomasses,darkmatter,beyondstandardmodel
5
1
0
2
I. INTRODUCTION

n
a
J Thediscoveryofverysmall,butnon-zeroneutrinomassesandtheexistenceofdarkmatter(DM)intheUniverse
may provide important information to guide us in the search for new physics beyond the standard model (SM).
9
Inrecentyearstheideatoincorporatebothphenomenainaunifiedframeworkhasreceivedmuchattention. And

] among the simplest realizations is the inert doublet model [1–3], which generates one-loop neutrino masses with
h
the DM being either an extra scalar-doublet or a Majorana fermion whose stability is protected by an exact Z
p 2
symmetry.
-
p Duetothesmallnessoftheneutrinomassscale,anumberofmodelswereproposedtogenerateneutrinomasses
e via higher loop processes, especially via 3-loop ones with the loop suppression (g2/16π2)3 10−13 (g being a
h electroweak-sized coupling) to naturally explain the large hierarchy m /v 10−13 (v being∼electroweak scale).
[ ν ∼
An earlier model [4] advocated by Krauss, Nasri and Trodden (KNT) extends the SM to include two charged

1 scalar singlets and a right-handed neutrino. Meanwhile, the model has an additional discrete symmetry, which
v makesneutrinomassesbefirstobtainedatthe3-looplevelviathenewparticleswiththemassesoforderofTeV.
0
Therefore, this model is phenomenologically interesting, and is well studied in the subsequent literatures [5–11].
2
Moreover, the generation of 3-loop neutrino masses also appear in the cocktail model [12], which adds to the SM
0
two scalar singlets (singly and doubly charged) and a scalar doublet.
2
0 In this paper, we present a new model by substituting a scalar triplet with hypercharge Y = 0 for a charged
. scalar singlet in the KNT model. Similarly, due to the additional Z symmetry and the field content of the
1 2
model, Majorana neutrino masses are also first generated at the 3-loop level, and the lightest Z -odd right-
0 2
5 handed Majorana fermion could be a DM candidate.
1 The paper is organized as follows: in Section 2 we describe the model, obtain the neutrino mass matrix, and
: calculate the DM annihilation processes. Various constraints on the model are analyzed numerically in Section
v
3. Then conclusions appear in Section 4.
i
X
r
a
II. A MODEL FOR NEUTRINO MASSES AND DARK MATTER
A. The model
In addition to SM fields, our model includes several right-handed Majorana fermions N , a charged SU(2)
iR L
singlet scalar S− and a triplet scalar ∆ with hypercharge Y =0
(cid:32) (cid:33)
√1 ∆0 ∆+
∆= 2 . (1)
∆− √1 ∆0
− 2
The number of N will be explained below. Moreover, we introduce a Z symmetry under which the new fields
iR 2
are all odd, whereas the SM fields are even. Given the symmetry and particle content of the model, the extra
∗Electronicaddress: [email protected]
2
Δ0 Δ0
W W W W W W
Hj− Hk− Hj− Hk− Hj− Hk−
να lα NiR lβ νβ να lα NiR lβ νβ να lα NiR lβ νβ
(a) (b) (c)
FIG. 1: Three-loop diagrams for radiative neutrino masses.
lagrangian will be
∆ = 1Tr(cid:104)(D ∆)2(cid:105)+(cid:0)D S−(cid:1)†DµS−+iN ∂N
L 2 µ µ iR (cid:54) iR
(cid:18) (cid:19)
1
m NT CN +g NT Cl S++h.c. V(∆,S−,Φ), (2)
− 2 Ni iR iR iα iR αR −
where C is the matrix of the charge conjugation and the covariant derivatives take the forms
D ∆ = ∂ ∆ ig (cid:2)Waτa,∆(cid:3) , (3)
µ µ − 2 µ
D S− = ∂ S−+ig(cid:48)B S−. (4)
µ µ µ
Here τa (a=1,2,3) is the Pauli matrix. The scalar potential of the new fields and the SM-like doublet Φ looks
like
V(∆,S−,Φ) = µ2 Φ†Φ+µ2S+S−+µ2 Tr[∆2]+λ (Φ†Φ)2+λ (S+S−)2
− H S ∆ 1 2
+λ (Tr[∆2])2+λ (Φ†Φ)(S+S−)+λ Tr[∆2](S+S−)
3 4 5
+λ6Φ†ΦTr[∆2]+(λ7Φ†∆Φ(cid:101)S++h.c.), (5)
where Φ˜ =iτ2Φ†.
As Z is exact, ∆ has no vacuum expectation value. After electroweak symmetry breaking, for λ = 0 the
2 7
charged Z -odd scalars ∆− and S− will mix (cid:54)
2
m2(∆−,S−)=(cid:18)2µ2∆+λ6v2 λ27v2 (cid:19), (6)
λ7v2 µ2 + λ4v2
2 S 2
where v 246 GeV is the vacuum expectation value of Φ. They will give rise to two charged mass eigenstates
≈
(cid:18)H− (cid:19) (cid:18) cosβ sinβ (cid:19)(cid:18)∆− (cid:19)
1 = . (7)
H− sinβ cosβ S−
2 −
Now the extra scalars are H−, H− and ∆0 with masses
1 2
(cid:113)
m m = cos2βm2 +sin2βm2 m . (8)
H1 ≤ ∆0 H1 H2 ≤ H2
B. Neutrino masses
Explicitly, the lagrangian in Eq. (2) breaks lepton number, and can generate a Majorana mass for the left-
handed neutrinos. However, The Z symmetry strictly forbids the generation of neutrino masses at either 1- or
2
2-loop order, and, therefore, the leading contributions to neutrino masses appear at 3-loop level shown in Fig. 1.
If the model has a single N , the neutrino mass matrix will predict two vanishing mass eigenvalues like the
R
case in Ref. [4] and contradict the neutrino oscillation data [6]. In order to solve the problem, one can add small
perturbations to the original mass matrix, add more scalars or right-handed Majorana fermions, and so on. In
this paper, we employ two right-handed fermions N (i=1,2) with m <m , which means that the Yukawa
iR N1 N2
couplings g can be complex and bring about three physical CP violation phases. However, in the following
iα
discussion, we leave aside the problem of CP violation for simplicity, so g takes real number.
iα
3
For the case of m > m m ,m ,m , it is appropriate to neglect the complicated contributions of
H2 ∆0 (cid:29) H1 Ni W
Figs. 1(b) and 1(c). Following the method in [7], we obtain the neutrino mass matrix elements arising from the
remaining Fig. 1(a)
(cid:88)
(M ) = g g m m I , (9)
ν αβ iα iβ α β i
i=1,2
where I is the three-loop integral
i
I = g4sin2(2β)mNi (cid:90) ∞ r dr (cid:110)12(cid:2)F (r,m2 ,m2 ) F (r,m2 ,m2 )(cid:3)2
i 6(16π2)3m4 r+m2 × 2 H1 H2 − 1 H1 H2
W 0 Ni
+(cid:2)G (r,m2 ,m2 ) G (r,m2 ,m2 )(cid:3)2 F (r,m2 ,m2 )(cid:2)5F (r,m2 ,m2 )
2 H1 H2 − 1 H1 H2 − 2 H1 H2 2 H1 H2
6F (r,m2 ,m2 ) G (r,m2 ,m2 )+G (r,m2 ,m2 )(cid:3)(cid:111). (10)
− 1 H1 H2 − 2 H1 H2 1 H1 H2
Here four integral functions have been introduced
(cid:90) 1 x(1 x)r+xm2
F (r,m2 ,m2 ) = dxln − H1 (m m ),
1 H1 H2 m2 − H1 → H2
0 W
(cid:90) 1 (1 x)(xr+m2 )+xm2
F (r,m2 ,m2 ) = dxln − W H1 (m m ),
2 H1 H2 m2 − H1 → H2
0 W
r+m2 (cid:90) 1 x(1 x)r+xm2
G (r,m2 ,m2 ) = H1 dx xln − H1 (m m ),
1 H1 H2 m2 m2 − H1 → H2
W 0 W
r m2 +m2 (cid:90) 1 (1 x)(xr+m2 )+xm2
G (r,m2 ,m2 ) = − W H1 dx xln − W H1 (m m ). (11)
2 H1 H2 m2 m2 − H1 → H2
W 0 W
The elements of the neutrino Majorana mass matrix M can be related to the mass eigenvalues
ν
M =UD UT with D =Diag(m ,m ,m ), (12)
ν ν ν 1 2 3
where U is the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) leptonic mixing matrix [13] parameterized by
 c c s c s e−iδ 1 0 0 
12 13 12 13 13
U = s12c23 c12s23s13eiδ c12c23 s12s23s13eiδ s23c13 0 eiα1/2 0  (13)
− − −
s12s23 c12c23s13eiδ c12s23 s12c23s13eiδ c23c13 0 0 eiα2/2
− − −
with c =cosθ , s =sinθ .
ij ij ij ij
Although the numerical results depend on the concrete choice of various parameters in the model, the above
neutrino mass matrix has the following special structure
  
a a 0 a a a
1e 2e 1e 1µ 1τ
Mν = a1µ a2µ 0a2e a2µ a2τ  (14)
a a 0 0 0 0
1τ 2τ
with a =g m √I . Therefore, the mass of the lightest neutrino is zero for Det(M )=0.
iα iα α i ν
C. Dark matter
WhenN isthelightestZ -oddstate,itisstableandcanbeaWIMPdarkmattercandidate. Form m ,
1 2 N2 (cid:29) N1
wecansafelyneglecttheeffectofN onN density. TheN numberdensitygetdepletedthroughtheannihilation
2 1 1
process N (p )N (p ) l+(p )l−(p ) via the t-channel and u-channel exchanges of H− . The amplitude for this
1 1 1 2 → α 3 β 4 1,2
process is
(cid:18) sin2β cos2β (cid:19)
= g∗ g + u(p )P u(p )v(p )P v(p )
Mαβ − 1α 1β t m2 t m2 4 L 2 1 R 3
− H1 − H2
(cid:18) sin2β cos2β (cid:19)
+g∗ g + u(p )P u(p )v(p )P v(p ), (15)
1α 1β u m2 u m2 4 L 1 2 R 3
− H1 − H2
4
where t = (p p )2 and u = (p p )2 are the Mandelstam variables corresponding to the t and u channels,
1 3 1 4
− −
respectively. After squaring, summing and averaging over the spin states, the total annihilation cross section in
the non-relativistic limit is given by
σv = (cid:80)(α,β)|g1∗αg1β|2m2 v2 (cid:34)sin4β(m4H1 +m4N1) + cos4β(m4H2 +m4N1)
rel 48π N1 rel (m2 +m2 )4 (m2 +m2 )4
H1 N1 H2 N1
(cid:35)
2sin2βcos2β(m2 m2 +m4 )
+ H1 H2 N1 , (16)
(m2 +m2 )2(m2 +m2 )2
H1 N1 H2 N1
where v is the relative velocity between the initial particles. Defining σv a+bv2 , we can approximately
rel rel ≡ rel
relate the dark matter relic abundance to the a and b variables by [14]
1.07 109 GeV−1 x 1
Ω h2 × F , (17)
N1 ≈ MP √g(cid:63)a+3(b a/4)/xF
−
where M =1.22 1019 GeV is the Planck scale, and g =86.25 is the number of relativistic degrees of freedom
P (cid:63)
×
at the freeze-out temperature x given by
F
(cid:34) (cid:114) (cid:35)
5 45 g M m (a+6b/x )
x =ln P N1 F . (18)
F 4 8 2π3 √g(cid:63)xF
Here g =2 is the number of degrees of freedom for the Majorana fermion dark matter.
III. EXPERIMENTAL CONSTRAINTS AND NUMERICAL RESULTS
Firstly, we summarize some relevant experimental data. A global fit to neutrino oscillation data gives [15]
s2 = 0.308 0.017,
12 ±
s2 = 0.437+0.033 (0.455+0.039),
23 −0.023 −0.031
s2 = 0.0234+0.0020 (0.0240+0.0019),
13 −0.0019 −0.0022
∆m2 = 7.54+0.26 10−5 eV2,
21 −0.22×
∆m2 = 2.43 0.06 (2.38 0.06) 10−3 eV2, (19)
| | ± ± ×
where the values (values in brackets) correspond to m <m <m (m <m <m ), i.e. normal mass spectrum
1 2 3 3 1 2
(inverted mass spectrum), and ∆m2 = m2 (m2 +m2)/2. As mentioned before, the lightest neutrino in the
3 − 2 1
model is massless, thus
m 0 (4.89 10−2) eV, m 8.68 10−3 (4.97 10−2) eV, m 4.97 10−2 (0) eV. (20)
1 2 3
(cid:39) × (cid:39) × × (cid:39) ×
The measured value of the relic density from WMAP [16] and Planck [17] is
Ωh2 =0.1199 0.0027. (21)
±
Moreover, the additional H− and N can mediate 1-loop lepton flavour violating (LFV) processes, such as
i i
l l γ (α=µ,τ, β =e,µ), and the branching ratios are
α β
→
(cid:12) (cid:12)2
Br(lα →lβγ) = 643παG2 (cid:12)(cid:12)(cid:12)(cid:12)(cid:88) gi∗αgiβ(cid:20)smin22βH(cid:18)mm22Ni(cid:19)+ cmos22βH(cid:18)mm22Ni(cid:19)(cid:21)(cid:12)(cid:12)(cid:12)(cid:12)
F (cid:12)i=1,2 H1 H1 H2 H2 (cid:12)
Br(l l ν ν¯ ), (22)
α β α β
× →
where α = e2/(4π) is the electromagnetic fine structure constant, G is the Fermi constant, and the function
F
H(x) is given by
1 6x+3x2+2x3 6x2lnx
H(x)= − − . (23)
6(1 x)4
−
5
6 0 0
c o s 2(cid:1) = 0 .1
5 0 0 c o s 2(cid:1) = 0 .2 5
c o s 2(cid:1) = 0 .4
m
4 0 0 N 2
]
V3 0 0
e
[G
m
2 0 0
m
H 1
1 0 0
0
1 2 3 4 5 6
m [ T e V ]
H 2
FIG.2: Theallowablevaluesofm andm satisfyingtheneutrinooscillationdata,LFVconstraintsandtheobserved
H1 N2
DM relic density.
The current experimental upper bounds of the LFV processes are [18, 19]
Br(µ eγ) < 5.7 10−13,
→ ×
Br(τ eγ) < 3.3 10−8,
→ ×
Br(τ µγ) < 4.4 10−8. (24)
→ ×
In addition, our model can generate effective four-lepton contact interactions at the 1-loop level, which can be
probed in e+e− collisions. Therefore, precise data from LEP will produce limits on the leptophilic dark matter.
The detail discussions can be found in Ref. [20, 21].
Now, we illustrate the allowed parameter space in the case of normal neutrino mass spectrum. The relevant
parameters in the model can be chosen as four particle masses m , m , a mixing angle β and six coupling
Hi Ni
constants g .
iα
Ingeneral,thestructureoftheneutrinomassmatrixinourmodelisinclinedtothehierarchyofg >g >g ,
ie iµ iτ
andtheobservedrelicabundanceimpliesthat(cid:80) g∗ g isoforder (1 10)for50GeV<m <200GeV.
(α,β)| 1α 1β| O − N1
Consequently, the Yukawa coupling constants could produce the large LFV branching ratios contradicting the
currentdata,especiallyforµ eγ. However,itisinterestingthatforsuitableparametervaluesthecontributions
→
of N and N in Eq. (22) can cancel out due to the opposite sign between g g and g g .
1 2 1e 1µ 2e 2µ
InFig.2,wekeepm =100GeV,g =0.9,anduseexperimentaldatainEq.(19),(21)and(24)todetermine
N1 1e
the allowable values of m and m according to m and cos2β. To guarantee the expression of the neutrino
H1 N2 H2
mass matrix in Eq. (9) only considering the contribution from Fig. 1(a), we pick m 1 TeV, which means
H2 ≥
that m in Eq. (8) is much larger than m , m and m . In this figure, one can find that the determination
∆0 N1 H1 W
of m weakly depends on m . For larger cos2β, such as cos2β >0.5, N is heavier than H− and can not be a
H1 H2 1 1
DM candidate. Meanwhile, for larger m , such as m > 7 TeV, g g and g g have the same sign, which
H2 H2 1e 1µ 2e 2µ
gives rise to unacceptable LFV.
In fact, six coupling constants g are also definite in the case of Fig. 2. Now we explicitly present their
iα
prototypical values in Fig. 3. Here we assume
m =5 TeV, m =1.5m , cos2β =0.25, (25)
H2 H1 N1
and m takes a suitable value to realize the cancellation in the decay µ eγ. As for m > 135 GeV, g is
N2 → N1 2µ
negative, so the cancellation disappears, which gives rise to unacceptable LFV. From the figure, we can give a
6
2
g [· 1 0 -2 ]
2 (cid:1)
g
2 e
1 g
1 e
g [· 1 0 -3 ]
2(cid:2)
i(cid:1) 0
g
g [· 1 0 -2 ]
1(cid:1)
-1
g [· 1 0 -3 ]
1(cid:2)
-2
5 0 6 0 7 0 8 0 9 0 1 0 0 1 1 0 1 2 0 1 3 0

m [G e V ]
N 1
FIG.3: AprototypicalchoiceoftheYukawacouplings. Notethatg andg aremultipliedby10−2and10−3,respectively.
iµ iτ
benchmark in the model. All input parameters are
m =100 GeV, m =350 GeV, m =150 GeV, m =5 TeV,
N1 N2 H1 H2
cos2β =0.25, g =0.909, g = 4.52 10−3, g = 1.29 10−3,
1e 1µ 1τ
− × − ×
g =1.01, g =1.35 10−2, g =5.99 10−4, (26)
2e 2µ 2τ
× ×
which leads to the neutrino oscillation data in Eq. (19), the DM relic density in Eq. (21) and the following LFV
results
Br(µ eγ) = 5.3 10−14,
→ ×
Br(τ eγ) = 1.8 10−12,
→ ×
Br(τ µγ) = 1.3 10−16. (27)
→ ×
They are consistent with the experimental bounds in Eq. (24).
Moreover,discussioninthecaseofinvertedneutrinomassspectrumissimilar,butthebiggercouplingconstants
lead to a much smaller viable parameter space.
IV. CONCLUSION
In this paper, we discussed an extension of the SM which includes two right-handed Majorana fermions N ,
i
a charged SU(2) singlet scalar S− and a triplet scalar ∆ with hypercharge Y = 0. Due to the additional
L
Z symmetry, the Z -odd fermion N could be a DM candidate and generate Majorana neutrino masses at the
2 2 1
3-loop level. Furthermore, the model predicts that the lightest neutrino is massless for the particular structure of
neutrino mass matrix. We also analyzed the constraints on the model coming from relevant experimental data,
and presented a prototypical allowed parameter choice, which favors the case of normal neutrino mass spectrum
and dark matter with m 50 135 GeV and a sizable Yukawa coupling constant g . It means that the DM and
1e
the charged scalar can be∼sear−ched for in future e+e− collisions.
Finally, we did not consider the problem of CP violation in the model. According to recent analyses [22, 23],
the best fit value of the Dirac CP violation phase is δ = 3π/2. Therefore, the coupling constants g can be
∼ iα
complex, and the model will possess more phenomenology.
7
Acknowledgments
This work is supported by the National Natural Science Foundation of China under Grant No. 11235005.
[1] E. Ma, Phys. Rev. D 73 (2006) 077301, arXiv:hep-ph/0601225.
[2] R. Barbieri, L.J. Hall and V.S. Rychkov, Phys. Rev. D 74 (2006) 015007, arXiv:hep-ph/0603188.
[3] L.L Honorez, E. Nezri, J.F. Oliver and M.H.G. Tytgat, JCAP 0702 (2007) 028, arXiv:hep-ph/0612275.
[4] L.M. Krauss, S. Nasri and M. Trodden, Phys. Rev. D 67 (2003) 085002, arXiv:hep-ph/0210389.
[5] E.A. Baltz and L. Bergstrom, Phys. Rev. D 67 (2003) 043516, arXiv:hep-ph/0211325.
[6] K. Cheung and O. Seto, Phys. Rev. D 69 (2004) 113009, arXiv:hep-ph/0403003.
[7] A. Ahriche and S. Nasri, JCAP 1307 (2013) 035, arXiv:1304.2055 [hep-ph].
[8] A. Ahriche, S. Nasri and R. Soualah, Phys. Rev. D 89 (2014) 095010, arXiv:1403.5694 [hep-ph].
[9] A. Ahriche, C.-S. Chen, K.L. McDonald and S. Nasri, Phys. Rev. D 90 (2014) 015024, arXiv:1404.2696 [hep-ph].
[10] A. Ahriche, K.L. McDonald and S. Nasri, J. High Energy Phys. 1410 (2014) 167, arXiv:1404.5917 [hep-ph].
[11] C.-S. Chen, K.L. McDonald and S. Nasri, Phys. Lett. B 734 (2014) 388, arXiv:1404.6033 [hep-ph].
[12] M. Gustafsson, J.M. No and M.A. Rivera, Phys. Rev. Lett. 110 (2013) 211802, arXiv:1212.4806 [hep-ph].
[13] B. Pontecorvo, Sov. Phys. JETP 26 (1968) 984 [Zh. Eksp. Teor. Fiz. 53 (1967) 1717]; Z. Maki, M. Nakagawa and S.
Sakata, Prog. Theor. Phys. 28 (1962) 870.
[14] Y. Bai and J. Berger, J. High Energy Phys. 1311 (2013) 171, arXiv:1308.0612 [hep-ph] .
[15] K.A. Olive, et al. [Particle Data Group], Chin. Phys. C 38 (2014) 090001.
[16] C.L. Bennett, et al. [WMAP Collaboration], Astrophys. J. Suppl. 208 (2013) 20, arXiv:1212.5225 [astro-ph.CO].
[17] P.A.R. Ade, et al. [Planck Collaboration], arXiv:1303.5076 [astro-ph.CO].
[18] J. Adam, et al. [MEG Collaboration], Phys. Rev. Lett. 110 (2013) 201801, arXiv:1303.0754 [hep-ex].
[19] B. Aubert, et al. [BaBar Collaboration], Phys. Rev. Lett. 104 (2010) 021802, arXiv:0908.2381 [hep-ex].
[20] P.J. Fox, R. Harnik, J. Kopp and Y. Tsai, Phys. Rev. D 84 (2011) 014028, arXiv:1103.0240 [hep-ph].
[21] A. Freitas and S. Westhoff, arXiv:1408.1959 [hep-ph].
[22] F. Capozzi, et al., Phys. Rev. D 89 (2014) 093018, arXiv:1312.2878 [hep-ph].
[23] M.C. Gonzalez-Garcia, M. Maltoni and T. Schwetz, arXiv:1409.5439 [hep-ph].