Can Zee-Babu model implemented with scalar dark matter explain both Fermi-LAT 130 GeV γ-ray excess and neutrino physics?
Department of Physics, Nagoya University
, Furo-cho, Chikusa-ku, Nagoya 464-8602,
School of Physics
, KIAS, 85 Hoegiro Dongdaemun-gu, Seoul 130-722,
We extend the Zee-Babu model for the neutrino masses and mixings by first incorporating a scalar dark matter X with Z2 symmetry and then X and a dark scalar with global U(1) symmetry. In the latter scenario the singly and doubly charged scalars that are new in the Zee-Babu model can explain the large annihilation cross section of a dark matter pair into two photons as hinted by the recent analysis of the Fermi -ray space telescope data. These new scalars can also enhance the B(H ), as the recent LHC results may suggest. The dark matter relic density can be explained. The direct detection rate of the dark matter is predicted to be about one order of magnitude down from the current experimental bound in the first scenario.
1 Introduction 2 3 4
B The annihilation cross section of XRXR
The Z2 model
2.1 Constraints on the potential
2.2 XX and Fermi-LAT 130 GeV -ray excess
2.3 Thermal relic density and direct detection rate
2.4 H , Z
3.2 Relic density in U(1)BL model
3.4 Implications for neutrino physics in both Z2 and U(1)BL model
Although it is well known that the dark matter (DM) constitutes about 27% of the total
mass density of the universe, i.e. DMh2 = 0.1199 0.0027 , its existence has only been
inferred from the gravitational interaction. And its nature is still unknown. If the DM
is weakly interacting massive particle (WIMP), it may reveal itself via non-gravitational
interactions, for example, by pair annihilation into ordinary standard model (SM) particles
including photon . In this case, the DM relic abundance is roughly related to the pair
annihilation cross section at freezeout, hvith, as
Refs. , more recently ref.  but with less significance, claim that the Fermi -ray
space telescope may have seen excess of the photons with E 130 GeV from the center of
the Milky Way compared with the background. Interpreting its origin as the annihilation
of a pair of DM particles, they could obtain the annihilation cross section to be about 4%
of that at freezeout:
hvi 0.04hvith 0.04 pb 1.2 1027cm3/s.
Since DM is electrically neutral, the pair annihilation process into photons occurs through
loop-induced diagrams. Naively we expect
So the observed value in (1.2) is rather large, and we may need new electrically charged
particles running inside the loop beyond the SM. Many new physics scenarios were speculated
within various CDM models by this observation .
The so-called Zee-Babu model  provides new charged scalars, h+, k++ at
electroweak scale, in addition to the SM particles. These new charged scalars carry two units
of lepton number and can generate Majorana neutrino masses via two-loop diagrams. The
diagrams are finite and calculable. The neutrino masses are naturally small without the
need to introduce the right-handed neutrinos for seesaw mechanism. One of the neutrinos
is predicted to be massless in this model. Both normal and inverse hierarchical pattern
of neutrino masses are allowed. The observed mixing pattern can also be accommodated.
The model parameters are strongly constrained by the neutrino mass and mixing data, the
radiative muon decay, e, and 3 decay .
It would be very interesting to see if the new charged particles in the Zee-Babu model
can participate in some other processes in a sector independent of neutrinos. In the first
part of this paper, we minimally extend the Zee-Babu model to incorporate the DM. In
the later part we will consider more extended model with global U(1)BL symmetry and
an additional scalar which breaks the global symmetry .
In the first scenario, we introduce a real scalar dark matter X with a discrete Z2
symmetry under which the dark matter transforms as X X in order to guarantee
its stability. The renormalizable interactions between the scalar DM X with the Higgs
field and the Zee-Babu scalar fields provide a Higgs portal between the SM sector and the
DM. We show that the Zee-Babu scalars and their interactions with the DM particle can
explain the DM relic density. The branching ratio of Higgs to two photons, B(H ),
can also be enhanced as implied by the recent LHC results . Although the charged
Zee-Babu scalars enhance the XX process, it turns out the the current experimental
constraints on their masses do not allow the annihilation cross section to reach (1.2).
In the extended scenario, we consider a complex scalar dark matter X and a dark scalar
. The global U(1) symmetry of the original Lagrangian is broken down to Z2 by
getting a vacuum expectation value (vev). We show that both the dark matter relic abundance
and the Fermi-LAT gamma-ray line signal can be accommodated via two mechanisms.
This paper is organized as follows. In section 2, we define our model by including the
scalar DM in the Zee-Babu model, and consider theoretical constraints on the scalar
potential. Then we study various DM phenomenology. We calculate the dark matter relic density
and the annihilation cross section hvi in our model. We also predict the cross section
for the DM and proton scattering and the branching ratio for the Higgs decay into two
photons, B(H ). And we consider the implication on the neutrino sector. In section 3,
we consider the DM phenomenology in the extended model. We conclude in section 4.
The Z2 model
We implement the Zee-Babu model for radiative generation of neutrino masses and
mixings, by including a real scalar DM X with Z2 symmetry X X. All the possible
renormalizable interactions involving the scalar fields are given by
+( hkhhk++ + h.c.)
+H (HH)2 + 4 X X4 + h(h+h)2 + k(k++k)2
+ 2 XhX2h+h + 2 XkX2k++k + hkh+hk++k.
Note that our model is similar to the model proposed by J. Cline . However we included
the interaction between the new charged scalar and the SM leptons that are allowed by
gauge symmetry, and thus the new charged scalar bosons are not stable and cause no
The original Zee-Babu model was focused on the neutrino physics, and the operators of
Higgs portal types were not discussed properly. It is clear that those Higgs portal operators
we include in the 2nd line of (2.3) can enhance H , without touching any other decay
rates of the SM Higgs boson, as long as h and k are heavy enough that the SM Higgs
decays into these new scalar bosons are kinematically forbidden.
Constraints on the potential
We require 2X , 2h and 2k to be positive. Otherwise the imposed Z2 symmetry X X
or the electromagnetic U(1) symmetry could be spontaneously broken down. Since the
masses of X, k++ and h+ have contributions from the electroweak symmetry breaking as
we obtain the conditions on the quartic couplings
We note that the above conditions are automatically satisfied if the couplings takes negative
values. In such a case, however, we also need to worry about the behavior of the Higgs
potential for large field values. For example, if we consider only the neutral Higgs field,
H,1 and the dark matter field, X, we get
1 1 1
V 4 H H4 + 4 X X4 + 4 HX H2X2,
for large field values of H and X. If the potential is to be bounded from below, every
eigenvalue of the square matrix of the couplings in (2.6) should be positive, whose condition is
This means that even if HX is negative, its absolute value should not be arbitrarily large
because H = m2H /2vH2 0.13 (mH 125 GeV) and X is bounded from above so as not
to generate the Landau pole. For example, the renormalization group running equations
(RGEs) of H , X and HX are given by
where the dots represents other contributions which are not important in the discussion.
The complete forms of the -functions of the quartic couplings are listed in appendix A.
The approximate solution for X in (2.8) shows that the Landau pole is generated at
the scale Q = QEW exp (1/H X (QEW)) (H = 18/162). If we take the electroweak scale
value of the Higgs quartic coupling to be X (QEW) 5, the cut-off scale should be around
1 TeV. The general condition for the bounded-from-below potential for large field values is that all the eigenvalues of the matrix
should be positive.
In the following discussion, we require that all scalar quartic couplings (i) be
perturbative up to some scale Q. To this end, we solve the one-loop RGEs of those quartic couplings
given in appendix A. For the moment, we do not include new Yukawa couplings defined in
eq. (2.2), and we adopt the criterion i(Q) < 4 in this analysis. In figure 1, the
perturbativity bounds are shown in the Xh(k)-Hh(k) plane. We take Q = 1, 3, 10 and 15 TeV, which
are denoted by the red curves from top to bottom. For other parameters, we fix Hh = Hk,
1We use the same notation with the Higgs doublet.
Xh = Xk, hk = HX = 0 and X = H ( 0.13). As explained above, a certain negative
value of Hk(h) may cause the instability of the Higgs potential. To avoid this, we set
h = Hh/(2H ) and k = 2Hk/(2H ) for Hk(h) < 0. For Hk(h) > 0, on the other hand,
h = k = H is taken. As we see from the plot, Xk(h) 7 11 is possible if Q = 1 TeV.
The theoretical arguments (2.5) and (2.7) restrict HX to lie roughly to the range,
(1.6, 0.6). Similarly, we have Hh(k) . 0.7 for mh+(k++) = 150 GeV.
where the amplitude-squared summed over the photon polarization is
Figure 2. Contour plot of hvi = (2, 1, 0.5, 0.2) 1027cm3/s (from above) in (mh+ , Xh) plane.
We set mX = 130 GeV, mH = 125 GeV, mk = 500 GeV and Xk = 5, HX = Hh = Hk = 0(0.33)
in the left (right) panel.
A1/2( ) = 2 h1 + (1 )f ( )i,
arcsin2 p1/ , ( 1)
f ( ) = 1 2
14 h log 11+1 ii , ( < 1).
Although the contribution of the doubly-charged Higgs k++ to hvi is 24 = 16 times
larger than that of the singly-charged Higgs h+ when their masses are similar to each other,
this option is ruled out by the recent LHC searches for the doubly-charged Higgs boson .
Depending on the decay channels, the 95% CL lower limit on the mass of the doubly-charged
Higgs boson is in the range, 204459 GeV. To be conservative, we set mk++ = 500 GeV.
In figure 2, we show a contour plot for the annihilation cross section into two photons:
hvi (2, 1, 0.5, 0.2) 1027cm3/s (from above) in the (mh+ , Xh) plane. We set mX =
130 GeV, mH = 125 GeV, mk++ = 500 GeV and Xk = 5, HX = Hh = Hk = 0 (0.33)
in the left (right) panel. We can see that by turning on the process, XX H , with
HX = 0.33 (right panel), we can reduce Xh to get hvi = 1 1027 cm3/s to explain
the Fermi-LAT gamma-ray line signal, but not significantly enough to push the cut-off
scale much higher than the electroweak scale. As we will see in the following section, the
hvi = 1 1027 cm3/s is not consistent with the current DM relic abundance.
Thermal relic density and direct detection rate
Contrary to J. Clines model , the DM relic density in our model is not necessarily
correlated with the hvi , since it is mainly determined by HX for relatively heavy scalars
(& 150 GeV). In this case the main DM annihilation channels are XX H SM particles,
where the SM particles are W +W , ZZ, bb, etc. As mh+(k++) becomes comparable with
mX , the XX h+h(k++k) modes can open, even in cases mX < mh+ (mk++ ) due to
the kinetic energy of X at freeze-out time. This can be seen in figure 3, where we show the
contour plot of DMh2 = 0.1199 (red lines) in the (Xh,HX ) plane for the choices mh+ =
150, 140, 130 GeV (shown in solid, dashed, dotted lines respectively). We fixed other
parameters to be mX = 130 GeV, mH = 125 GeV, mk = 500 GeV, Xk = 5, Hh = Hk = 0.5.
For mh+ = 130 GeV, the annihilation mode XX h+h dominates even for very small
coupling Xh (the red dotted line). The black vertical lines are the constant contour lines of
hvi = 0.21027cm3/s. We can see that the maximum value for the Fermi-LAT
gammaray line signal which is consistent with the relic density is hvi = 0.2 1027cm3/s when
mh+ = 150 GeV. This cross section is smaller than the required value in (1.2) by factor 6.
Figure 4 shows the cross section of dark matter scattering off proton, p, as a function
of HX (red solid line) and p = 1.8 109 pb line (black dashed line) above which is
excluded by LUX  at 90% C.L. This cross section is determined basically only by HX
at tree level by the SM Higgs exchange, when we fix mX = 130 GeV. We can see that
HX . 0.06 to satisfy the LUX upper bound.
In this scenario the decay width of H , Z can be modified, whereas other Higgs
decay widths are intact. The decay width of H  is given by
(H ) = 64e2m3mvH2H HhA0(h+ ) + 4HkA0(k++ )
where i = 4mi2/m2H (i = f, W, h+, k++). For the H Z we adapted the formulas in
refs. [69, 70] for our model:
2Ncf Qf (Tf3L 2Qf xW )
+ Hk(c2W 1)vH2 A0(k++ , k++ )
where. The loop functions are
where the function f is defined in (2.13) and
H aL m h + =1130 GeV, m k ++ =500 GeV
In figure 5, we show contour plots for constant (H )/(H )SM (black solid
lines) and (H Z)/(H Z)SM (black dashed lines) in the (Hh, Hk) plane. For
this plot we set mh+ = 130 (150) GeV for the left (right) panel and fixed mk++ = 500 GeV.
The shaded regions are disfavored by (2.5) (blue) and by (2.7) (yellow). The ratios depend
basically only on the coupling constants Hh and Hk as well as the masses mh+ and mk++ .
And the ratios are not necessarily correlated with the hvi which are controlled by Xh
and Xk. We can conclude that
0.54 . (H )/(H )SM . 1.45 (1.35)
0.91 . (H Z)/(H Z)SM . 1.11 (1.08)
for the left (right) panel. That is, the H channel can be enhanced (reduced)
significantly, whereas the H Z channel can change only upto 10%.
As we have seen in the previous section, the simplest extension of Zee-Babu model to
incorporate dark matter with Z2 symmetry, although very predictive, has difficulty in fully
explaining the Fermi-LAT gamma-line anomaly. In this section we consider a next minimal
model where we may solve the problem. We further extend the model by introducing
U(1)BL symmetry and additional complex scalar to break the global symmetry [49, 50].
Then the model Lagrangian (2.3) is modified as
where we also replaced the real scalar dark matter X in (2.3) with the complex scalar field.
The charge assignments of scalar fields are given as follows:
where is the Goldstone boson associated with the spontaneous breaking of global
U(1)BL. For convenience we also rotated the field X
where cH cos H , sH sin H , with H mixing angle, and we take H1 as the
SMlike Higgs field. Then mass matrix can be written in terms of mass eigenvalues mi2 of
Hi(i = 1, 2):
m21c2H + m22s2H (m22 m21)cH sH !
(m22 m21)cH sH m21s2H + m22c2H
In the scalar potential we have 22 parameters in total. We can trade some of those
parameters for masses,
As will be discussed later, we may need parameter region where v( 106 GeV) is very large
but m2 is at electroweak scale. From (3.5), we get m22/(2v2) and H m21/(2vH2 ).
The vacuum stability condition similar to (2.7) gives a constraint on H:
X = XR+2iXI .
where mR(I) is the mass of XR(I). For simplicity we take XR as the dark matter candidate
from now on. We can also express H , , H in terms of masses mi2(i = 1, 2) and mixing
angle H , then we take the 22 free parameters as
vH ( 246 GeV),
m1( 125 GeV),
where two values, vH and m1, have been measured as written in the parentheses.
In this section we will see that we can obtain dark matter annihilation cross section into
two photons, XRXR , large enough to explain the Fermi-LAT 130 GeV -line excess.
There are two mechanisms to enhance the annihilation cross section in this model:
H2resonance and large v. In these cases, since the SM Higgs, H1, contribution is small for
small mixing angle H , we consider only the contribution of H2 assuming H = 0 (or H2 =
). Allowing non-vanishing H would only increase the allowed region of parameter space.
Then we obtain the annihilation cross section times relative velocity for XRXR ,
(2 X + X v)v X Qi2i[1 if (i)]
s m2 + im
+ X Qi2Xi[1 if (i)] ,
where Qi is electric charge of i(= h+, k++), i = 4mi2/s and is total decay width of .
Since vrel 103 1, we can approximate s = 4m2R/(1 vr2el/4) 4m2R. When H = 0,
the H2(= ) can decay into two Goldstone bosons () or into two photons with partial
(2 X + X v)2 uuvt1 4m2R(I) ,
X Qi2i[1 if (i)] .
As mentioned above, figure 6 shows the two enhancement mechanisms for XRXR :
the left panel for the -resonance and the right panel for the large v. For these plots we set
the parameters: mR = 130, mI = 2000, mh+ = 300, mk++ = 500 (GeV), h = k = 0.1,
X = Xh = Xk = 0.01, v = 1000 (GeV) for the left plot and m = 600 (GeV) for
the right plot. We can obtain the large annihilation cross section required to explain
Fermi-LAT gamma-line data either near the resonance, m 2mR (left panel) or at
large v (right panel). These behaviors can be understood easily from (3.14). In either
of these cases only the 1st term in (3.14) gives large enhancement. The slope on the
right of the resonance peak (the left panel of figure 6) is steeper than that on the left
because, when m > 260 GeV, new annihilation channel XRXR opens and the decay
width of increases leading to decreasing the annihilation cross section. In the right
panel of figure 6, the dip near v 104 GeV occurs because there is cancellation between
2 X = (m2R mI2)/(2v) and X v terms for positive X .
The two mechanisms can also be seen in figure 7. This figure shows a contour plot
of v(XRXR ) = 0.04 (pb) in (v, X )-plane. We set mR = 130, mI = 1000,
mh+ = 1000, mk++ = 1000, m = 260 (GeV), h = k = Xh = Xk = 0.01 for red
lines (-resonance). And we take mI = 1000, mh+ = 300, mk++ = 500, m = 600 (GeV),
h = k = 0.1, Xh = Xk = 0.01 for blue line (large v). The red (blue) lines represent
the -resonance (large v) solution for Fermi-LAT anomaly. In the -resonance region, for
the negative (positive) X the two values 2 X = (m2R mI2)/(2v) and X v which
appear in the 1st term of (3.14) have the same (opposite) sign and their contributions are
constructive (destructive). As a result for positive X (solid red line), there is cancellation
between the two terms, and larger value of v is required for a given X . For large v case,
the result does not depend on the sign of X because the X v term dominates. And the
solid and dashed blue lines overlap each other in figure 7. For X larger than about 0.1 the
decay width ( XRXR) becomes too large to enhance the annihilation cross section.
Relic density in U(1)BL model
Now we need to check whether the large enhancement in XRXR signal is consistent
with the observed relic density DMh2 = 0.1199 0.0027. To obtain the current relic
density the DM annihilation cross section at decoupling time should be approximately
(assuming S-wave annihilation)
hvith 3 1026 cm3/s 1 pb,
from (1.1). The major difference between the Z2 model and the U(1)BL model is
that the latter model has additional annihilation channel, i.e., XRXR and
exchange s-channel diagrams compared with the former one. The S-wave contribution
to v(XRXR ) is shown in the appendix. The Goldstone boson mode becomes
dominant especially when v is not very large , i.e. v . 103 GeV. And it makes the dark
matter phenomenology very different from the one without it. For example, in Z2 model
we need the annihilation channel XX h+h(k++k) large enough to obtain the
current relic density. In U(1)BL model, however, the annihilation into Goldstone bosons are
sometimes large enough to explain the relic density.2
To see the relevant parameter space satisfying both the Fermi-LAT 130 GeV
gammaline anomaly and the correct relic density, we consider the -resonance and large v cases
discussed above separately. Figure 8 shows contours of v(XRXR ) = 0.04 (pb)
(solid line) and XR h2 0.12 (dashed line) for X > 0 when the resonance condition
m = 2mR is satisfied. The parameters are chosen as m = 2mR = 260 GeV, mI = mh+ =
mk++ = 1 TeV, h = k = Xh = Xk = 0.01. We can see there are intersection points
of the two lines where both Fermi-LAT anomaly and the relic density can be explained.
For the parameters we have chosen the contribution of XRXR to the relic density is
almost 100%. This implies there is wide region of allowed parameter space satisfying both
2The dark sector can be in thermal equilibrium with the SM plasma in the early universe even with very
small mixing H 108 . And our analysis with H = 0 can be thought of as a good approximation
of more realistic case of non-zero but small H .
observables, since other annihilation channels XX h+h(k++k) are also available
when they are kinematically allowed. Typically TeV scalar v gives too large XRXR
annihilation cross section resulting in too small relic density. For the positive X case,
however, there is also cancellation between terms in v(XRXR ) as in v(XRXR
). Both cancellations are effective when the condition, X v2 = (mI2m2R)/2, is satisfied.
This explains the intersection point occurs on the diagonal straight line determined by the
above condition. This allows large relic density even near TeV v.
Figure 9 shows the same contours for X < 0. In this case as we have seen in figure 7
that TeV scale v can explain Fermi-LAT gamma-line. However this value of v gives
too large DM annihilation cross section at the decoupling time (when XRXR is
dominant) and too small relic density. So somehow we need to decouple the XRXR
so that we need larger v. We can do it, for example, by assuming h+(k++) are
v(XRXR ) 0.04 pb and XR h2 0.12 as can be seen in figure 9. The two
lines meet at rather large v( 105 GeV) as expected. For other parameters we chose3
m 2mR = 260 GeV, mI = 200 GeV, h = k = Xh = Xk = 0.01. The pattern
of the relic density contour requires some explanation. The annihilation cross section for
Figure 9. The same plot with figure 8 for X < 0. We also take the -resonance condition,
2mR = m = 260 GeV. See the text for other parameters. The region to the right of the dashed
line gives DMh2 > 0.12.
where is the total decay width of . The non-vanishing partial decay widths of for
the parameters we chose are ( ), ( XRXR) and ( ). In figure 10
they are plotted as a function of v for X = 107. On the vertical part of the relic
density contour in figure 9 near v 104.4 GeV, the ( ) dominates and also
2 X X v. For this v we approximately get
which is independent of X . Around v 107.2 GeV, ( XRXR) and ( )
dominate despite high phase space suppression in XRXR, and 2 X X v.
As X increases, ( XRXR) becomes more important than ( ) as can be
seen from (3.15) and (3.16). The (almost) vertical part for this v region is due to partial
cancellation of the factor (2 X +X v)2 in the numerator of (3.21) and the same factor
in the cross term of ( XRXR) and ( ) in the denominator. As X grows
even larger, only ( XRXR) term dominates and v(XRXR ) 1/2X v4, which
gives the slanted part of the contour line.
We can also obtain simultaneous solutions when is off-resonance using large v.
Figure 11 shows an example of this case. In this case, if we have only XRXR
channel for relic density, the resulting XR h2 is too large for v & 1 TeV. To get the
Figure 11. The same plot with figure 8 corresponding to large v solution. See the text for the
parameters used in this figure.
correct relic density by increasing the DM pair annihilation cross section at freeze-out we
allowed XRXR h+h channel. Then we can get a solution as can be seen in figure 11.
The region enclosed by two dashed lines over-closes the universe. For this plot, we chose
mR = 130 GeV, m = mI = 1 TeV, mh+ = 150 GeV, mk++ = 500 GeV, h = 0.001, and
h = k = Xh = Xk = 0.01. Note that we take h = 0.001 so that the solid red
line representing v(XRXR ) = 0.04 pb and dashed line representing h2 = 0.1199
overlap with each other. To show that this choice of s is possible in general, we take a
point on the overlapped lines, e.g., v = 106.43 GeV and X = 0.001. Then we can get
h = 0.001, h = 0.01 as a solution from figure 12.
When there is no mixing between and h the decay width H1 is the same with
that of the SM. This means that we can enhance the XRXR without affecting the
SM H1 rate. When the mixing angle H is non-vanishing the h+ and/or k++ can
contribute to H1 through one-loop process. Since this effect was already discussed
in section 2.4, we do not discuss it further.
Other cosmological implications of U(1)BL model: Neff , topological
defects, self-interacting dark matter
Weinberg  showed that Goldstone bosons can play the role of dark radiation and
contribute to the effective number of neutrinos Neff . If Goldstone bosons go out of
equilibrium when the temperature is above the mass of muons but below that of all other
SM particles, we get Neff = 0.39. The condition for this to happen can be roughly
estimated by setting the collision rate nhvi( ) is equal to the Hubble expansion
rate H T 2/mpl. The is dominated by the s-channel scalar exchange diagram.
It is mediated by an operator
which is generated by terms 1/v, HvvH h and m/vH in (3.1). Since
n T 3 and a derivative yields a factor T in thermal averaging, nhvi H gives
After being decoupled from thermal plasma, the energy from muon annihilation heats up
neutrinos but not . This enhances the neutrino temperature relative to the temperature
of the Goldstone boson. From the entropy conservation before and after muon annihilation,
we get T /T = (57/43)1/3. The Neff is equal to the energy fraction of a Goldstone
boson relative to a single neutrino: Neff = T4/(7/4T4) = 4/7(43/57)4/3 = 0.39. Actually
WMAP9 and ground-based observations  give Neff = 3.89 0.67 and Planck, the
WMAP9 polarization and ground-based observations  give Neff = 3.36 0.34, both
at the 68% confidence leve, suggesting possible deviation from the SM prediction although
the errors are large. For example, with H = 0.005(0.0001) and mH = 125 GeV,
the dark scalar with mass 500 MeV (70 MeV) can satisfy the condition. With this light
dark scalar, obviously the resonance solution for Fermi-LAT gamma line is not applicable.
From (3.5) we can see with H = 0.0001 and m2 = 70 MeV, if we take sH = 0.3 which
is consistent with Higgs invisible decay width at the LHC, we get = 2 108 and
v = 190 TeV, which is consistent with both (2.7) and figure 11 for the Fermi-LAT gamma
line. As analyzed in section 3.2, we can easily find the dark sector parameters to explain
the current relic abundance.
At the early universe when the U(1)BL symmetry is broken, cosmic strings can be
produced. Although the energy per unit length of a long straight string from global
symmetry breaking diverges logarithmically, the energy per length of two anti-parallel strings
is finite . The contribution of cosmic strings to the total energy budget of the universe
is given by 
where mpl 1019 GeV is Planck mass and v2 ln(vd) width d the characteristic
distance between strings. The constraint /m 2pl . 106  is easily satisfied for the values
of v . 107 GeV taken in our scenario.
After U(1)BL symmetry breaking, our model has remnant discrete Z2 symmetry. If
this Z2 symmetry is spontaneously broken, domain walls can be formed and dominate the
energy density of the universe, causing domain wall problem .
If inflation occurs after the U(1)BL symmetry breaking, these topological defects
are diluted and do not make any problems. However, recent BICEP2 observation 
suggests that the inflation scale may be much higher than the U(1)BL breaking scale.
And we cannot resort to inflation to solve the domain wall problem, if they are produced.
However, if we assume the Goldstone boson is massless, the potential along the -direction
is flat (Z2 is exact), the domain walls are not produced, because the Z2 symmetry is
not spontaneously broken. In more general, the Goldstone bosons get masses from higher
dimensional operators suppressed by Planck mass which are generated by quantum gravity.
This can break Z2 symmetry explicitly and there is no domain wall problem, although
some higher dimensional operators should be suppressed to guarantee the longevity of
dark matters. Our scenario is different from the axion case where the axion masses are
induced by QCD instantons making the axion potential have discrete symmetry that is
The self-interacting dark matters  with long-range force have received much
interest because they can solve core/cusp problem  and too big to fail problem .
The current bound on self-interacting dark matters is given by
where T = R d(1 cos )d/d is the transfer cross section. If the above cross section
is close to the bound, the dark matters can solve the problems. The Goldstone boson
which is assumed to be massless or very light couples to the dark matter with interaction
suppressed by v as can be seen from the rotation in (3.3). Although the Goldstone boson
could be thought to mediate long-range force, actually the force scales not as 1/r but as
1/r3 due to its pseudo-scalar nature , and cannot contribute to the self-interactions
of dark matters much. However, if the scalar is light (sub-GeV scale) it can mediate
long-range Yukawa interactions making our dark matter self-interacting. It is interesting
to see that light can enhance both Neff and the self-interactions.
Since the new particles X (in both Z2 and U(1)BL model) and (in U(1)BL model)
beyond the Zee-Babu scalars h+, k++ do not couple directly to lepton doublets, the
neutrino phenomenology is the same with the original Zee-Babu model and no further
constraints are imposed by X or at least in the tree level. It is well known the Zee-Babu
model is strongly constrained by charged lepton flavor violating (LFV) processes such as
e or . The -parameter is constrained to be less than about 500 GeV
to make the scalar potential stable .
The most recent constraints on Zee-Babu model were studied in . In the analysis
they used the updated values of 13 and e and the updated LHC results. They found
that the neutrino oscillation data and low energy experiments are compatible with masses
of the extra charged scalar mass bounds from the LHC.
We have considered two scenarios which minimally extended the Zee-Babu model [44
46]. In the first scenario we introduced a real scalar dark matter X with Z2 symmetry:
X X. If the scalar dark matter X has a mass around 130 GeV, the annihilation cross
section, hvi(XX ), can be enhanced by the contribution of the singly- and/or
doublycharged Zee-Babu scalars. If we also want to explain the dark matter relic abundance,
however, we get at most hvi 0.2 1027cm3/s, which is about factor 6 smaller than
the required value to explain the Fermi-LAT gamma-ray line signal.
We have shown that the present constraint on the couplings Xk and Xh which
mix the dark matter and charged Higgs is not so strong and they can enhance the
annihilation cross section of XX large enough to accommodate the recent hint. On
the other hand the couplings which involve the SM Higgs H are strongly constrained by
the theoretical considerations in the Higgs potential and the observations of dark matter
relic density and dark matter direct detections. The upper bound on the HX coupling
is about 0.06 which comes from the dark matter direct detection experiments. For the
Hh, Hk which mix the SM Higgs and the new charged Higgs, the theoretical bound
becomes more important. If we require the absolute stability of the dark matter by the
Z2 symmetry X X and the absence of charge breaking, we get the upper bound
of Hh, Hk to be about 0.7 for the charged Higgs mass around 150 GeV. To evade the
unbounded-from-below Higgs potential we need to have Hh, Hk & 1.6. With these
constraints the B(H (Z)) can be enhanced up to 1.5 (1.1) or suppressed down to
0.5 (0.9) with respect to that in the SM. The neutrino sector cannot be described by the
Zee-Babu model only, and there should be additional contributions to the neutrino masses
and mixings such as dimension-5 Weinberg operator from type-I seesaw mechanism.
In the second scenario, we introduced two complex scalar fields X and with global
U(1)BL symmetry. After gets vev, v, the U(1)BL symmetry is broken down to Z2
symmetry. The lighter component of X, which we take to be the real part, XR, is stable
due to the remnant Z2 symmetry and can be a dark matter candidate. Even in the extreme
case where we do not consider the mixing of the dark scalar and the standard model Higgs
scalar (H = 0), we showed that the dark matter relic abundance and the Fermi-LAT
gamma-ray line signal can be accommodated in two parameter regions: resonance region
(m = 2mR) and large v( 106 107GeV) region. Since there is no mixing, there is no
correlation with H and direct detection scattering of dark matter off the proton. In
addition the neutrino sector need not be modified contrary to the first scenario.
We thank Wan-Il Park for useful discussions. This work is partly supported by NRF
Research Grant 2012R1A2A1A01006053 (PK, SB).
Here, we give the renormalization group equation and the one-loop functions of the
X = 1612 h182X + 22HX + 2Xh + 2Xki,
Hh = 162 12H Hh + 8hHh + 2hkHk + HX Xh + 3g14
Hk = 162 12H Hk + 8kHk + 2hkHh + HX Xk + 12g14
(3g22 + g12) 6yt2
where (4m2R m2)2 should be replaced by (4m2R m2)2 + 2m2 in the denominator when
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