Can Zee-Babu model implemented with scalar dark matter explain both Fermi-LAT 130 GeV γ-ray excess and neutrino physics?

Journal of High Energy Physics, Sep 2014

Abstract We extend the Zee-Babu model for the neutrino masses and mixings by first incorporating a scalar dark matter X with Z 2 symmetry and then X and a dark scalar φ 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 γ-ray space telescope data. These new scalars can also enhance the B(H → γγ), 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.

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Can Zee-Babu model implemented with scalar dark matter explain both Fermi-LAT 130 GeV γ-ray excess and neutrino physics?

Seungwon Baek 1 P. Ko 1 Hiroshi Okada 1 Eibun Senaha 0 1 0 Department of Physics, Nagoya University , Furo-cho, Chikusa-ku, Nagoya 464-8602, Japan 1 School of Physics , KIAS, 85 Hoegiro Dongdaemun-gu, Seoul 130-722, Korea 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 [1], 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 [2]. In this case, the DM relic abundance is roughly related to the pair annihilation cross section at freezeout, hvith, as 3 1027cm3/s Refs. [1316], more recently ref. [17] 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 105. 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 [1843]. The so-called Zee-Babu model [4446] 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 [47]. 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 [50]. 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 [6365]. 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 [50]. 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.) 1 +H (HH)2 + 4 X X4 + h(h+h)2 + k(k++k)2 1 1 + 2 XhX2h+h + 2 XkX2k++k + hkh+hk++k. Note that our model is similar to the model proposed by J. Cline [19]. 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 problem. 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, 2 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 2 , 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 [66]. 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 [19], 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, HX0.03 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 [67] 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 [68] 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++ ) m2k++ 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: H = 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 . X = 2X = 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 decay width (2 X + X v)2 uuvt1 4m2R(I) , 32m m2 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 [50], 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 [51]. 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 [52] 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 1. 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 [5355] give Neff = 3.89 0.67 and Planck, the WMAP9 polarization and ground-based observations [5659] 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 [60]. The contribution of cosmic strings to the total energy budget of the universe is given by [60] where mpl 1019 GeV is Planck mass and v2 ln(vd) width d the characteristic distance between strings. The constraint /m 2pl . 106 [61] 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 [62]. 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 [3] 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 broken spontaneously. The self-interacting dark matters [510] with long-range force have received much interest because they can solve core/cusp problem [4] and too big to fail problem [11]. The current bound on self-interacting dark matters is given by mDM v=10km . 35cm2/g, 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 [12], 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 [47]. The -parameter is constrained to be less than about 500 GeV to make the scalar potential stable [46]. The most recent constraints on Zee-Babu model were studied in [48]. 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. Conclusions 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. Acknowledgments 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 quartic couplings: X = 1612 h182X + 22HX + 2Xh + 2Xki, 1 Hh = 162 12H Hh + 8hHh + 2hkHk + HX Xh + 3g14 1 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 2mR m. Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

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Seungwon Baek, P. Ko, Hiroshi Okada, Eibun Senaha. Can Zee-Babu model implemented with scalar dark matter explain both Fermi-LAT 130 GeV γ-ray excess and neutrino physics?, Journal of High Energy Physics, 2014, 153, DOI: 10.1007/JHEP09(2014)153