Table Of ContentRenormalizable Model for Neutrino Mass, Dark Matter, Muon g 2
−
and 750 GeV Diphoton Excess
Hiroshi Okada1,2 and Kei Yagyu3
1School of Physics, KIAS, Seoul 130-722, Korea
2Physics Division, National Center for Theoretical Sciences, Hsinchu, Taiwan 300
3School of Physics and Astronomy, University of Southampton,
6
1 Southampton, SO17 1BJ, United Kingdom
0
2
We discuss a possibility to explain the 750 GeV diphoton excess observed at the LHC in
n
a a three-loop neutrino mass model which has a similar structure to the model by Krauss,
J
Nasri and Trodden. Tiny neutrino masses are naturally generated by the loop effect of new
9
1
particles with their couplings and masses to be of order 0.1-1 and TeV, respectively. The
] lightest right-handed neutrino, which runs in the three-loop diagram, can be a dark matter
h
p candidate. Inaddition,thedeviationinthemeasuredvalueofthemuonanomalousmagnetic
-
p
momentfromitspredictioninthestandardmodelcanbecompensatedbyone-loopdiagrams
e
h with exotic multi-charged leptons and scalar bosons. For the diphoton event, an additional
[
isospin singlet real scalar field plays the role to explain the excess by taking its mass of
1
v 750 GeV, where it is produced from the gluon fusion production via the mixing with the
8
3 standardmodellike Higgsboson. We find thatthe crosssectionofthe diphotonprocesscan
0
5 be obtained to be a few fb level by taking the masses of new charged particles to be about
0
375 GeV and related coupling constants to be order 1.
.
1
0
6 PACS numbers:
1
:
v
i I. INTRODUCTION
X
r
a
In December 2015, the both ATLAS and CMS groups have reported the existence of the excess
at around 750 GeV in the diphoton distribution at the Large Hadron Collider (LHC) with the
collision energy of 13 TeV. Thelocal significance of this excess is about3.6σ at ATLAS [1]with the
integrated luminosity of 3.2 fb 1 and about 2.6σ at CMS [2] with the integrated luminosity of 2.6
−
fb 1. The detailed properties of the diphoton excess was summarized, e.g., in Ref. [3], where the
−
best fit value of the width of the new resonance is about 45 GeV, and the estimated cross section
of the diphoton signature is 10 3 fb at ATLAS and 6 3 fb at CMS. If this excess is confirmed
± ±
by future data, it suggests the existence of a new particle which gives the direct evidence of a new
physics beyond the standard model (SM).
2
The simplest way to explain this excess is to consider an extension of the SM by adding extra
isospinscalarmultiplets suchasasinglet, adoubletand/oratriplet. However, itisdifficulttogeta
sufficientcross section toexplain theexcess as mentioned in theabove insuchasimpleextension of
the SM. For example, if we consider the CP-conserving two Higgs doublet models (THDMs) [4–8],
and take the masses of the additional CP-even H and CP-odd A Higgs bosons to be 750 GeV,
then the cross section of pp H/A γγ is typically three order smaller than the required
→ →
value [4]. Therefore, we need to further introduce additional sources to get an enhancement of the
production cross section and/or the branching fraction to the diphoton mode, e.g., by introducing
multi-charged scalar particles [4, 6] and vector-like fermions [7]. In Refs. [8], the diphoton excess
has been discussed in supersymmetric models.
In this paper, we discuss a scenario to naturally introduce multi-charged particles to get an
enhancement of the branching fraction. Namely, we consider a radiative neutrino mass model
in which multi-charged particles play a role not only to increase the branching fraction but also
to explain the smallness of neutrino masses and the anomaly of the muon anomalous magnetic
moment. A dark matter (DM) candidate can also successfully be involved as a part of the model.
There are a few papers discussing the diphoton excess within radiative neutrino mass models [9].
In particular, we discuss a new three loop neutrino mass model1 whose structure is similar to
the model by Krauss, Nasri and Trodden in 2003 [10], because the three loop suppression factor
1/(16π2)3 is a suitable amount to reproduce the measured neutrino masses, i.e., (0.1) eV, by
O
order 0.1-1 couplings and TeV scale masses of new particles. In our model, an additional isospin
real singlet scalar field can explain the diphoton excess, where it is produced from the gluon fusion
process through the mixing with the SM-like Higgs boson.
The plan of the paper is as follows. In Sec. II, we define our model, and give the Lagrangian
for the lepton sector and the scalar potential. In Sec. III, we discuss the neutrino masses, the
phenomenology of DM including the relic abundance and direct search experiments, and new
contributions to the muon g 2. The diphoton excess is discussed in Sec. IV. Our conclusion is
−
summarized in Sec. V.
1 Othervariations of threeloop neutrinomass models have also been proposed in Refs. [11–16].
3
Lepton Fields Scalar Fields
Li ei La =(L−−a,L−−−a)T E−−a Na Φ κ++ S++ Σ
L R 5/2 R
(SU(2) ,U(1) ) (2, 1/2) (1, 1) (2, 5/2) (1, 2) (1,0) (2,1/2) (1,2) (1,2) (1,0)
L Y
− − − −
Z2 + + + + +
− − − −
TABLE I: Particle contents and charge assignments under SU(2)L U(1)Y Z2. The superscripts i and a
× ×
denote the flavor of the SM fermions and the exotic fermions with i=1-3 and a=1-N , respectively.
E
II. THE MODEL
Our model is described by the SM gauge symmetry SU(2) U(1) and an additional discrete
L Y
×
symmetry Z which is assumed to be unbroken. This Z symmetry is introduced to avoid tree
2 2
level contributions to neutrino masses and to enclose the three-loop diagram as shown in Fig. 1.
Because of the Z symmetry, the stability of the lightest neutral Z odd particle is guaranteed,
2 2
and thus it can be a candidate of DM.
The particle contents are shown in Table I, where Li and ei are the SM left-handed lepton
L R
doublets and lepton singlets with the flavor of i (i =1-3). In addition, we add the N flavor of
E
the vector like lepton doublets (singlets) La = (L a,L a)T (E a) with the hypercharge
5/2 −− −−− −−
Y = 5/2 ( 2) and the right-handed neutrinos Na (a = 1-N ). The scalar sector is composed of
− − R E
one isospin doublet field Φ with Y = 1/2 and two complex (one real) isospin singlet scalar fields
κ++ and S++ with Y = 2 (Σ with Y = 0). The doublet and the real singlet scalar fields are
parameterized by
G+
Φ = , Σ = σ0+v , (1)
v+φ0+iG0 σ
√2
where v and v are the vacuum expectation values (VEVs) of doublet and singlet scalar fields,
σ
respectively, and G+ (G0) denotes the Nambu-Goldstone boson which is absorbed into the longi-
tudinal component of the W (Z) boson. TheFermi constant G is given by the usual relation, i.e.,
F
G = 1/(√2v2) with v 246 GeV. The singlet VEV v does not contribute to the electroweak
F ≃ σ
symmetry breaking. We note that the shift v v does not change any physical quantities,
σ → σ′
because its impact can be absorbed by the redefinition of the parameters in the Lagrangian. We
thus take v = 0 in the following discussion to make some expressions to be a simple form.
σ
4
The most general Lagrangian for the lepton fields is given by
1
= MaNacNa +Ma(La ) (La ) +MaE aE a+h.c.
−Llep 2 N R R L 5/2 L 5/2 R E L−− R−−
+yi Li Φei +yab(La ) Φ˜E b+yab(La ) Φ˜ E b+h.c.
SM L R 1 5/2 L R−− 2 5/2 R L−−
+gab(La ) (La ) Σ+gab(E a) E bΣ+gabNacNb Σ+h.c.
L 5/2 L 5/2 R E −− L R−− N R R
+hijeicej κ+++habNaE bκ+++habNacE bκ++ +fiaLi (La ) S+++h.c., (2)
0 R R 1 R L−− 2 R R−− L 5/2 R
where Φ˜ = iσ Φ . We can take the diagonal form of the invariant masses Ma, Ma and Ma for the
2 ∗ N L E
vector likeleptons La , Ea andright-handedneutrinosNa, respectively, withoutlossof generality.
5/2 R
The SM leptons L and e are taken to be the mass eigenstates, so that the Yukawa coupling yi
L R SM
is given by the diagonal form. For simplicity, we assume that all the above parameters are real.
The most general Higgs potential is given by
V(Φ,κ++,S++,Σ)= V (Φ,Σ)+µ2(κ++κ )+µ2(S++S )
HSM κ −− S −−
+A Σ(κ++κ )+A Σ(S++S )
Σκ −− ΣS −−
+λ (Φ Φ)(κ++κ )+λ (Φ Φ)(S++S )+λ Σ2(κ++κ )+λ Σ2(S++S )
Φκ † −− ΦS † −− Σκ −− ΣS −−
+λ (κ++κ )2+λ (S++S )2+λ (κ++κ )(S++S )
κ −− S −− κS −− −−
+[λ (κ++S )(κ++S )+h.c.], (3)
0 −− −−
where the complex phase of the λ parameter can be absorbed by rephasing the scalar fields. The
0
squared masses of the doubly-charged scalar bosons S and κ are given by
±± ±±
v2 v2
m2 = µ2 + λ , m2 = µ2 + λ . (4)
κ±± κ 2 Φκ S±± S 2 ΦS
In Eq. (3), the V part is given as the same form as in the Higgs singlet model (HSM) involving
HSM
Φ and Σ as
V (Φ,Σ) =µ2(Φ Φ)+λ(Φ Φ)2+A (Φ Φ)Σ+λ (Φ Φ)Σ2
HSM Φ † † ΦΣ † ΦΣ †
+t Σ+µ2Σ2+A Σ3+λ Σ4. (5)
Σ Σ Σ Σ
Two CP-even scalar states φ0 from the doublet and s0 from the singlet are mixed with each other
via the mixing angle α defined as
σ0 cosα sinα H
= − . (6)
φ0 sinα cosα h
5
FIG. 1: Three-loop neutrino mass diagram. Particles indicated by the red color are Z2 odd, where Eα−−a
and E−−c denote the mass eigenstates for the doubly-charged exotic leptons. The subscripts α and β (run
β
over 1 and 2) label two mass eigenstates for each flavor.
We define h as the SM-like Higgs boson with the mass of about 125 GeV which is identified as the
discovered Higgs boson at the LHC. The detailed expressions for the masses of the CP-even Higgs
bosons and the mixing angle α in terms of the potential parameters are given, e.g., in Ref. [17].
The masses of the exotic charged leptons are obtained from two sources, i.e., the invariant mass
terms M and M and the Yukawa interaction terms y and y . The mass of the triply-charged
E L 1 2
leptons L a is simply given by Ma. For the doubly-charged leptons, there is a mixing between
−−− L
L a and E a through the y and y terms. The mass matrix is given assuming yab = yab =
−− −− 1 2 1 2
ya = diag(y1,...,yNE) by
E E E
Ma Ma E a Ma 0 E a
= (E a,L a) E D −− = (E a,E a) E1 1−− , (7)
Lmass − −− −− Ma MaL a − 1−− 2−− 0 Ma E a
D L −− E2 2−−
whereMa = v ya. Themasseigenstates Ea andEa aredefinedbytheorthogonaltransformation:
D √2 E 1 2
E a cosθ sinθ E a
−− = a − a 1−− . (8)
L a sinθ cosθ E a
−− a a 2−−
The mass eigenvalues (Ma Ma ) and the mixing angles θ are given by
E1 ≤ E2 a
1
Ma = Ma +Ma (Ma Ma)2+4(Ma)2 , (9)
E1,2 2 E L∓ E − L D
(cid:18) q (cid:19)
2Ma
tan2θ = D . (10)
a Ma Ma
E − L
6
III. NEUTRINO MASS, DARK MATTER, MUON g 2
−
A. Neutrino Mass
The leading contribution to the active neutrino mass matrix m is given at three-loop level
ν
as shown in Fig. 1. One- and two-loop diagrams which have been systematically classified in
Refs. [18, 19] are absent in our setup. The three-loop diagram is computed as follows
λ
0
(m ) =
ν ij (16π2)3M2
max
3 2
× cαβfia∗ sin2θaMEaαhb1aMNb hb1c sin2θcMEcβfjc∗F(rEαa,rNb,rEβc,rS±±,rκ±±), (11)
a,b,c=1α,β=1
X X
where we define rX ≡ (mX/Mmax)2 with Mmax = Max(MEαa,MEβc,MNb,mS±±,mκ±±) and mX is
the mass of a particle X, and c = 1( 1) for α = β(α = β). The three loop function F is given
αβ
− 6
by
1 1
F(rEαa,rNb,rEβc,rS±±,rκ±±)= dx1dy1dz1δ(1−x1−y1−z1)z2 z
Z0 1 − 1
1 1 1 1
dx dy dz δ(1 x y z ) dx dy dz δ(1 x y z ) , (12)
× 2 2 2 − 2− 2− 2 z2 z 3 3 3 − 3− 3− 3 ∆
Z0 2 − 2 Z0 3
where
∆3 = x3rEa +y3rS±± +z3∆2, (13)
α
with ∆ = x2rNb +y2∆1+z2rκ±±, and ∆ = xrEβc +yrS±± +zrκ±±. (14)
2 − z2 z 1 − z2 z
2 − 2 1 − 1
1
The interval of the integrals in Eq. (12) for all the variables is from 0 to 1, i.e., dxdydz =
0
1 1 1
dx dy dz. Typical values of F with r = (0.1,1) are (1). Let us estimate mRagnitudes of
0 0 0 i O
RcoupliRngs anRd masses to reproduce the magnitude of neutrino masses, i.e., the order of 0.1 eV. For
simplicity, when we take Mmax = MEαa ∼ MEβc ∼ MNb, the neutrino masses are approximately
expressed as
M M
(m ) max K (0.1 eV) 106 max K , (15)
ν ij ∼ (16π2)3 × ij ∼ × × v × ij
(cid:18) (cid:19)
3
with K = fia hbahbcfjc , (16)
ij ∗ 1 1 ∗
a,b,c=1
X
where we assume λ F = (1). Therefore, in the range of M = v-10v, the magnitude of the
0 max
× O
mixing factor K is required to be (10 7-10 6).
ij − −
O
7
B. Dark Matter
Assuming that the right-handed neutrino N1 is the lightest among all the Z odd particles, N1
R 2 R
looses its decay modes into any other lighter particles, and then it becomes stable. We thus can
regard N1 as the DM candidate in our model. The annihilation cross section is then calculated as
R
π 2π 1 4m2
σv dθsinθ dφ (N1N1 AB)2 1 fin, (17)
rel ≈ 128π2s|M R R → | s − s
0 0
Z Z
where m is the mass of the final state particle. In the above expression, (N1N1 AB)2 is
fin |M R R → |
the squared amplitude for the following two body to two body processes:
(N1N1 AB)2 = (N1N1 κ++κ )2 + (N1N1 ff¯)2+ (N1N1 ZZ)2
|M R R → | |M R R → −− | |M R R → | |M R R → |
+ (N1N1 W+W )2 + (N1N1 hh)2 + (N1N1 HH)2, (18)
|M R R → − | |M R R → | |M R R → |
The first annihilation process N1N1 κ++κ happens through the t- and u-channels of the Ea
R R → −− α
mediation, where the doubly-charged scalar bosons κ decays into the same sign dilepton via the
±±
Yukawa coupling h . The squared amplitude of the N1N1 κ++κ process is given by
0 R R → −−
NE
(N1N1 κ++κ )2 = h1a 2tr (/p M1)X (/p +M1)X , (19)
|M R R → −− | | | 2 − N a 1 N a†
Xa=1 h i
/p +k/ +Ma /p +k/ +Ma
X = cos2θ − 1 1 E1 + − 1 2 E1 +(cosθ sinθ , Ma Ma ), (20)
a a t (Ma )2 u (Ma )2 a → a E1 → E2
" − E1 − E1 #
f
where s, t and u are the Mandelstam variables, N is the color factor, and (p ,p ) and (k ,k ) are
c 1 2 1 2
theinitialandthefinalstate momenta, respectively. Inthisexpression, wetake hab hab = hab for
1 2
≡
simplicity. The other cross sections are given through the mixing of α via the s-channel mediation
8
of h and H by
g11m s c 2 1 1 2
(N1N1 ff¯)2 = 16Nf N f α α
|M R R → | c v s m2 +im Γ − s m2 +im Γ
(cid:18) (cid:19) (cid:12) − h h h − H H H(cid:12)
(cid:12) (cid:12)
(p p ) (M1)2 (cid:12)(k k ) m2 , (cid:12)(21)
× 1· 2 − N (cid:12) 1· 2 − f (cid:12)
(cid:2)g11m2s c 2 (cid:3)(cid:2) 1 (cid:3) 1 2
(N1N1 ZZ)2 = 8 N Z α α
|M R R → | v s m2 +im Γ − s m2 +im Γ
(cid:18) (cid:19) (cid:12) − h h h − H H H(cid:12)
× (p1·p2)−(MN1(cid:12)(cid:12)(cid:12))2 2+ (k1m·4k2)2 , (cid:12)(cid:12)(cid:12) (22)
(cid:20) Z (cid:21)
(cid:2) g11m2 s c 2 (cid:3) 1 1 2
(N1N1 W+W )2 = 16 N W α α
|M R R → − | v s m2 +im Γ − s m2 +im Γ
(cid:18) (cid:19) (cid:12) − h h h − H H H(cid:12)
× (p1·p2)−(MN1)2(cid:12)(cid:12)(cid:12) 2+ (k1m·4k2)2 , (cid:12)(cid:12)(cid:12) (23)
(cid:20) W (cid:21)
(cid:2) (cid:3) 2
(N1N1 hh)2 = 2 g11 2 sαλhhh cαλHhh
|M R R → | N s m2 +im Γ − s m2 +im Γ
(cid:12) − h h h − H H H(cid:12)
(cid:0) (cid:1) (cid:12) (cid:12)
(p p(cid:12) ) (M1)2 , (cid:12) (24)
× 1· (cid:12)2 − N (cid:12)
2
(N1N1 HH)2 = 2 (cid:2)g11 2 sαλhHH(cid:3) cαλHHH
|M R R → | N s m2 +im Γ − s m2 +im Γ
(cid:12) − h h h − H H H(cid:12)
(cid:0) (cid:1) (cid:12) (cid:12)
(p p(cid:12) ) (M1)2 , (cid:12) (25)
× 1· (cid:12)2 − N (cid:12)
(cid:2) (cid:3)
where we use the short-hand notations of c cosα and s sinα. The dimensionful λ
α ≡ α ≡ ϕiϕjϕk
couplings (ϕ = h or H) are defined by the coefficient of the scalar trilinear vertex in the
i,j,k
potential. We note that the s-wave contribution to σv vanishes due to the Majorana property
rel
of the DM. To reproduce the observed relic density, the cross section given in Eq. (17) should be
inside the following region
σv = (1.78-1.97) 10 9 GeV 2, (26)
rel − −
×
at the 2σ level [20].
We also consider the spin independent scattering cross section with a neutron that is induced
via the tree level diagram with the Higgs boson h and H exchange. The formula is given by
C2 m2M1 2 g11c s 2 1 1 2
σn = n N N α α + , (27)
SI π m +M1 v −m2 m2
(cid:18) n N(cid:19) (cid:18) (cid:19) (cid:18) h H(cid:19)
where the neutron mass is m 0.939 GeV and the factor C 0.2872 is determined by the lattice
n
≃ ≃
simulation. The latest upper bound is reported by the LUX experiment that suggests σn . 10 45
SI −
cm for the DM mass of about 100 GeV with the 90 % C.L. [21].
9
C. Muon g 2
−
The muon anomalous magnetic moment (muon g 2) is one of the most promising low energy
−
observables which suggest the existence of new physics beyond the SM. This is because there is the
morethan 3σ deviation in theSMprediction fromthe experimentalvalue measuredat Brookhaven
National Laboratory. The difference ∆a aexp aSM has been calculated in Ref. [22] as
µ ≡ µ − µ
∆a = (29.0 9.0) 10 10. (28)
µ −
± ×
This shows the 3.2σ deviation in the SM prediction.
In our model, two diagrams contribute to ∆a , where L a-S a and ℓ -κ a with ℓ being
µ −−− −− − −− −
the SM lepton are running in the loop. These contributions are calculated by
m2 NE 3 m2 2 Ma2 3 2
∆a µ fµa 2 G S±± + G L h¯µi 2 (29)
µ ≃ 16π2 | | Ma2 Ma2 m2 m2 − | 0 | 3m2
(a=1 (cid:20) L (cid:18) L (cid:19) S±± (cid:18) S±±(cid:19)(cid:21) i=1 κ±±)
X X
where h¯ij = hij(2hij) for i= j(i = j), and
0 0 0 6
1 6x+3x2+2x3 6x2lnx
G(x) = − − . (30)
(1 x)4
−
We can see that the contribution from the κ loop gives the negative value which is undesired to
±±
explain the muon g 2 anomaly. We thus neglect the κ loop contribution that can be realized
±±
−
by taking h¯µi fµa.
0 ≪
D. A set of solution
Here, we show a set of the solution to give the sizable amount of ∆a , i.e., 2.0 10 9 .
µ −
×
∆a . 3.8 10 9, the non-relativistic cross section to satisfy the observed relic density σv =
µ − rel
×
(1.78-1.97) 10 9 GeV 2, and to satisfy the constraint of the direct detection σn . 10 45, where
× − − SI −
we conservatively take the constraint of the direct detection for all the mass region of DM. By
taking the number of the flavor N = 3, we find the following benchmark parameter sets:
E
mκ±± = 375 GeV, mS±± = 377 GeV, ME1 = 375 GeV, ME2 = 380 GeV,
M1 = 478 GeV, M2,3 = 556 GeV, m = 750 GeV, Γ = 2.40 GeV,
N N H H
3 3
fµa 2 = 3.042, h1a 2 = 0.7212, g11 =1.06 10 3, sinθ = 0.141, sinα = 0.1, (31)
| | | | N × − E −
a=1 a=1
X X
where M = Ma , θ = θ for a = 1-3, and we take λ = λ = 0. The triply-charged
E1,2 E1,2 E a hhh Hhh
lepton mass M is given about 375 GeV from the above inputs. The values for three parameters
L
10
m , sinα and Γ are favored for the discussion of the 750 GeV diphoton signature which will be
H H
discussed in the next section. The other SM parameters are fixed as follows
Γ = 4.1 MeV, m = 125.5 GeV, v = 246 GeV, m = 80.4 GeV, m = 91.2 GeV,
h h W Z
m = 173 GeV, m = 4.18 GeV. (32)
t b
From the benchmark set, we obtain the following results
∆a = 3.54 10 9, σv = 1.87 10 9 GeV 2. (33)
µ − rel − −
× ×
IV. DIPHOTON EXCESS
We discuss how we can reproduce the diphoton excess at around 750 GeV at the LHC. In our
model, the additional CP-even Higgs boson H plays the role to explain this excess via the gluon
fusion production process by taking its mass of 750 GeV. The cross section σ of the diphoton
γγ
channel is expressed by using the narrow width approximation as follows
σ σ(gg H γγ) σ(gg H) (H γγ). (34)
γγ
≡ → → ≃ → ×B →
Non-zero production cross section σ(gg H) of the gluon fusion process is given through the
→
mixing α with the SM-like Higgs boson h defined in Eq. (6) as
σ(gg H)= sin2α σ(gg h ), (35)
SM
→ × →
where h denotes the SM Higgs boson, and σ(gg h ) does its gluon fusion cross section in
SM SM
→
which the mass of h here is fixed to be 750 GeV in order to derive the cross section for H.
SM
From [23], we obtain σ(gg h ) 736 fb at the collision energy of 13 TeV.
SM
→ ≃
Next, we discuss the decays of H and h to figure out the branching fraction of (H γγ) and
B →
the signal strength µ of pp h XY modes for h. The latter quantity becomes important
XY → →
to set a constraint on the parameter space. In particular, when we consider the enhancement of
(H γγ), this could also significantly modify the event rates of h for various channels. The
B →
definition of µ is given by
XY
µ = σ(gg h) (h XY). (36)
XY → ×B →
The decay rates of with = h or H and = ff¯, W+W , ZZ or gg are given by
′ ′ −
H → PP H PP
Γ( )= ξ2Γ(h ), (37)
′ SM ′
H → PP H → PP
