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Figure 6. Comparison of reaction densities (Yann,p) for annihilation and the decay. Before EWSB we show the s-channel f f + X® process (black dashed) and t-channel ®f + Xf (red) and also the decay contribution (black doted), while after EWSB we plot the density for the decay (solid black) only. The solid black line (hidden beneath the red line before EWSB) shows both contributions from the annihilation and the decay together. In the left panel we choose mg = 500 GeV, mx = 100 GeV, while in the right, we choose me = 100 GeV, mx = 20 GeV. In both cases we have a = 107!6 GeV~*, 6 = 0.1. The vertical dashed-dotted lines shows x corresponding to EWSB.

Figure 6 Comparison of reaction densities (Yann,p) for annihilation and the decay. Before EWSB we show the s-channel f f + X® process (black dashed) and t-channel ®f + Xf (red) and also the decay contribution (black doted), while after EWSB we plot the density for the decay (solid black) only. The solid black line (hidden beneath the red line before EWSB) shows both contributions from the annihilation and the decay together. In the left panel we choose mg = 500 GeV, mx = 100 GeV, while in the right, we choose me = 100 GeV, mx = 20 GeV. In both cases we have a = 107!6 GeV~*, 6 = 0.1. The vertical dashed-dotted lines shows x corresponding to EWSB.

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Table 1. Charges of the new particles under SU(3). x SU(2)z x U(1)y x Z symmetry.
Table 1. Charges of the new particles under SU(3). x SU(2)z x U(1)y x Z symmetry.
Figure 1. Top: the contours show allowed range of the vev vg and mass of the decoupled heavy scalar mg as a function of Aq and Ag y for fixed Ag = 0.1. Bottom left: Agy as a functions of m2 for fixed values of Ay. Bottom right: sin(2a) as a functions of mz for fixed values of Az. Decoupling of the dark sector is clearly seen in the limit of large me.
Figure 1. Top: the contours show allowed range of the vev vg and mass of the decoupled heavy scalar mg as a function of Aq and Ag y for fixed Ag = 0.1. Bottom left: Agy as a functions of m2 for fixed values of Ay. Bottom right: sin(2a) as a functions of mz for fixed values of Az. Decoupling of the dark sector is clearly seen in the limit of large me.
where ellipsis denote interactions of the SM Higgs boson h with other SM components that are not relevant here. We note here that we necessarily assume A > mo, otherwise higher dimensional operators (neglected in this work) would appear in the scalar potential. We also adopt the following notations hereafter: a(@) = ote) B=%.
where ellipsis denote interactions of the SM Higgs boson h with other SM components that are not relevant here. We note here that we necessarily assume A > mo, otherwise higher dimensional operators (neglected in this work) would appear in the scalar potential. We also adopt the following notations hereafter: a(@) = ote) B=%.
Figure 3. Annihilations via ¢-channel (top) and s-channel (bottom) leading to the production of VDM X before EWSB. The diagrams with Goldstone bosons identically vanish, leaving only two diagrams with SM fermions.
Figure 3. Annihilations via ¢-channel (top) and s-channel (bottom) leading to the production of VDM X before EWSB. The diagrams with Goldstone bosons identically vanish, leaving only two diagrams with SM fermions.
Figure 4. Annihilations via t-channel (left panel) and s-channel (right panel) leading to the production of X after EWSB.
Figure 4. Annihilations via t-channel (left panel) and s-channel (right panel) leading to the production of X after EWSB.
Following eq. (3.8) and eq. (3.11) the total yield due to decay and due to annihilation can be written as:
Following eq. (3.8) and eq. (3.11) the total yield due to decay and due to annihilation can be written as:
Figure 5. Top panels: comparison of the yield at Tew in the massive case versus that in the massless case with respect to the reheat temperature, where the red, green and blue curves correspond to the massive cases while the black curve represents the massless case. Bottom panel: Variation of AY/Ym=o with the reheat temperature for different choices of the masses shown in different colors. In all the plots “massless” refers to > m = 0.
Figure 5. Top panels: comparison of the yield at Tew in the massive case versus that in the massless case with respect to the reheat temperature, where the red, green and blue curves correspond to the massive cases while the black curve represents the massless case. Bottom panel: Variation of AY/Ym=o with the reheat temperature for different choices of the masses shown in different colors. In all the plots “massless” refers to > m = 0.
Figure 7. Total reaction density over the whole range of temperature before and after EWSB taking into account both annihilation and decay before EWSB, but only decay after EWSB. We consider all SM states to be massless before EWSB, while dark sector particles are massive. The dashed-dotted vertical lines correspond to tpw = Tew for each choice of DM mass. In the left panel we show densities for different choices of ms = {50, 100,300, 500} GeV (in red, green, blue, black, respectively) for fixed r = = 7x = = 0.2, while in the right one they are shown for different r = {0.2,0.4,0.6,0.8} (in red, prean; blue, black, respectively) for fixed me = 100 GeV. In both cases we choose a = 107'!4 GeV~!, 8 = 0.1.
Figure 7. Total reaction density over the whole range of temperature before and after EWSB taking into account both annihilation and decay before EWSB, but only decay after EWSB. We consider all SM states to be massless before EWSB, while dark sector particles are massive. The dashed-dotted vertical lines correspond to tpw = Tew for each choice of DM mass. In the left panel we show densities for different choices of ms = {50, 100,300, 500} GeV (in red, green, blue, black, respectively) for fixed r = = 7x = = 0.2, while in the right one they are shown for different r = {0.2,0.4,0.6,0.8} (in red, prean; blue, black, respectively) for fixed me = 100 GeV. In both cases we choose a = 107'!4 GeV~!, 8 = 0.1.
Figure 8. Top left: evolution of the DM yield Yx for parameters which imply correct relic DM abundance as a function of « = mx/T obtained by solving eq. (4.1). Here different colored curves = indicate different choices of r = “* = {0.1,0.2,0.4,0.6} in red, green, blue and black respectively for a fixed mp = 100 GeV,a = 107!® GeV~', 6 = 0.1 and reheat temperature Try = 10° GeV. The vertical dashed-dotted lines vomreapen to “ew = mx/Tew for different values of mx. Top right: same as top left but with ms = 500 GeV. Bottom left: relic density allowed parameter space in terms of mx — a where different colored contours correspond to me = 500 GeV (red) and me = 100 GeV (blue) for a fixed 6 = a = 0.1 and reheat temperature Try = 10° GeV. Bottom right: same as bottom left but with Try = 10° GeV.
Figure 8. Top left: evolution of the DM yield Yx for parameters which imply correct relic DM abundance as a function of « = mx/T obtained by solving eq. (4.1). Here different colored curves = indicate different choices of r = “* = {0.1,0.2,0.4,0.6} in red, green, blue and black respectively for a fixed mp = 100 GeV,a = 107!® GeV~', 6 = 0.1 and reheat temperature Try = 10° GeV. The vertical dashed-dotted lines vomreapen to “ew = mx/Tew for different values of mx. Top right: same as top left but with ms = 500 GeV. Bottom left: relic density allowed parameter space in terms of mx — a where different colored contours correspond to me = 500 GeV (red) and me = 100 GeV (blue) for a fixed 6 = a = 0.1 and reheat temperature Try = 10° GeV. Bottom right: same as bottom left but with Try = 10° GeV.
Figure 9. Top panel: comparison of reaction densities due to annihilation (colorful lines) and decay (shown in black) after EWSB. Contributions of two annihilation channels are shown: h,® > X, Z via Z boson mediation in t-channel and h,® + X,7(Z) via ® mediation in s-channel. All states are assumed to be massive. The amplitudes for h,® — X,7(Z) channel are proportional to the portal coupling Az7@ that is being varied in the plots. The choice of parameters is specified in the headings. Bottom panel: Decay branching fraction of ® > X+(Z) as function of r= ™* Me
Figure 9. Top panel: comparison of reaction densities due to annihilation (colorful lines) and decay (shown in black) after EWSB. Contributions of two annihilation channels are shown: h,® > X, Z via Z boson mediation in t-channel and h,® + X,7(Z) via ® mediation in s-channel. All states are assumed to be massive. The amplitudes for h,® — X,7(Z) channel are proportional to the portal coupling Az7@ that is being varied in the plots. The choice of parameters is specified in the headings. Bottom panel: Decay branching fraction of ® > X+(Z) as function of r= ™* Me
Figure 10. Top left: DM yield as a function of x = mx/T for a reheat temperature Tay = 10 TeV considering both annihilation and decay to be active before EWSB, while keeping only decay after EWSB. We choose me = 100 GeV and 6 = 0.1 while for each value of mx, a@ is adjusted to reproduce the central value of the measured relic abundance. Top right: same as top left but with Tru = 1 TeV. In both of the plots the vertical dashed lines denote EWSB (apw). Bottom left: relic density allowed parameter space in mx — a plane for Tay = 10 TeV and a adjusted to reproduce the DM abundance. Bottom right: same as bottom left but with Tru = 1 TeV.
Figure 10. Top left: DM yield as a function of x = mx/T for a reheat temperature Tay = 10 TeV considering both annihilation and decay to be active before EWSB, while keeping only decay after EWSB. We choose me = 100 GeV and 6 = 0.1 while for each value of mx, a@ is adjusted to reproduce the central value of the measured relic abundance. Top right: same as top left but with Tru = 1 TeV. In both of the plots the vertical dashed lines denote EWSB (apw). Bottom left: relic density allowed parameter space in mx — a plane for Tay = 10 TeV and a adjusted to reproduce the DM abundance. Bottom right: same as bottom left but with Tru = 1 TeV.
Figure 11. Top panel: DM yield as a function of temperature for different choices of the reheat temperature Try : {10°,10?,10,1} TeV shown in red, green, blue and black curves respectively. Here we have considered both annihilation and decay channels before EWSB where SM particles are massless while the dark particles are massive. In the post-EWSB scenario we have considered only the decays with massive states. The parameters chosen correspond to right relic abundance. n each of these plots the vertical dashed-dotted lines denote EWSB (aw). Middle left: relic density allowed parameter space in terms of a — Try where different colored contours correspond to different mx for a fixed mse = 100 GeV. Middle right: same as middle left panel, but for a different fixed me = 500 GeV. Bottom left: same as middle panel but for different choices of 8 : {0.001, 0.1, 0.3, 0.5, 1.0, 3.0, 5.0} for me = 100 GeV and mx = 10 GeV. Bottom right: same as bottom left with me = 500 GeV and mx = 100 GeV
Figure 11. Top panel: DM yield as a function of temperature for different choices of the reheat temperature Try : {10°,10?,10,1} TeV shown in red, green, blue and black curves respectively. Here we have considered both annihilation and decay channels before EWSB where SM particles are massless while the dark particles are massive. In the post-EWSB scenario we have considered only the decays with massive states. The parameters chosen correspond to right relic abundance. n each of these plots the vertical dashed-dotted lines denote EWSB (aw). Middle left: relic density allowed parameter space in terms of a — Try where different colored contours correspond to different mx for a fixed mse = 100 GeV. Middle right: same as middle left panel, but for a different fixed me = 500 GeV. Bottom left: same as middle panel but for different choices of 8 : {0.001, 0.1, 0.3, 0.5, 1.0, 3.0, 5.0} for me = 100 GeV and mx = 10 GeV. Bottom right: same as bottom left with me = 500 GeV and mx = 100 GeV
Figure 13. The contours show different decay lengths of ® for particular choices of the parameters that can produce right relic abundance for the DM. On left panel, we choose mg = 100 GeV, while on the right panel we show the case for me = 500 GeV. The shaded region at the bottom is excluded by BBN constraint Te S Tgpn ~ 1 sec. Figure 13. The contours show different decay lengths of ® for particular choices of the parameters
Figure 13. The contours show different decay lengths of ® for particular choices of the parameters that can produce right relic abundance for the DM. On left panel, we choose mg = 100 GeV, while on the right panel we show the case for me = 500 GeV. The shaded region at the bottom is excluded by BBN constraint Te S Tgpn ~ 1 sec. Figure 13. The contours show different decay lengths of ® for particular choices of the parameters
Figure 14. £7; + jets signature that the model can produce at the collider. the phenomenology of the model is severely constrained. First of all, note that there is no XX — SM vertex, therefore no elastic scattering of the DM against nuclei is possible. So, this model easily avoids stringent constraints from non-observation of DM scattering at direct search experiments. DM can only scatter off nuclei inelastically with ® in the final state as shown in left panel of figure 12. Since ms > mx, hence such an inelastic scattering is forbidden even if the mass difference 6m = me —mx 2 O(100) MeV [89]. On the other hand, due to the presence of © — X — y vertex, the DM pair annihilation may give rise to monochromatic X-ray line (right panel of figure 12) but such photon flux will be hugely suppressed by 1/A? and can not account for the, say, galactic-center gamma ray excess as observed. Hence this model in its freeze-in realization of DM can not be probed rom either direct nor indirect DM search experiments.
Figure 14. £7; + jets signature that the model can produce at the collider. the phenomenology of the model is severely constrained. First of all, note that there is no XX — SM vertex, therefore no elastic scattering of the DM against nuclei is possible. So, this model easily avoids stringent constraints from non-observation of DM scattering at direct search experiments. DM can only scatter off nuclei inelastically with ® in the final state as shown in left panel of figure 12. Since ms > mx, hence such an inelastic scattering is forbidden even if the mass difference 6m = me —mx 2 O(100) MeV [89]. On the other hand, due to the presence of © — X — y vertex, the DM pair annihilation may give rise to monochromatic X-ray line (right panel of figure 12) but such photon flux will be hugely suppressed by 1/A? and can not account for the, say, galactic-center gamma ray excess as observed. Hence this model in its freeze-in realization of DM can not be probed rom either direct nor indirect DM search experiments.
Table 3. Relevant DM-SM interaction vertices and vertex factors along with SM propagators. All momenta are assumed flowing towards the vertices. Here @ stands for SM leptons and @ stands for the SM quark generations. ye@) is the hypercharge for left and right-chiral leptons (quarks): Yy =-1,Y¢ =-1, Yg = -2; YP = 1/3, Yg = 4/3, Y¢ = —2/3. Table 3. Relevant DM-SM interaction vertices and vertex factors along with SM propagators. All
Table 3. Relevant DM-SM interaction vertices and vertex factors along with SM propagators. All momenta are assumed flowing towards the vertices. Here @ stands for SM leptons and @ stands for the SM quark generations. ye@) is the hypercharge for left and right-chiral leptons (quarks): Yy =-1,Y¢ =-1, Yg = -2; YP = 1/3, Yg = 4/3, Y¢ = —2/3. Table 3. Relevant DM-SM interaction vertices and vertex factors along with SM propagators. All
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