Large scalar multiplet dark matter in the high-mass region

We study two models of scalar dark matter from “large” electroweak multiplets with isospin $5/2$ ($n=6$ members) and $7/2$ ($n=8$), whose scalar potentials preserve a $Z_2$ symmetry. Because of large annihilation cross sections due to electroweak interactions, these scalars can constitute all the dark matter only for masses in the multi-TeV range. For such high masses, Sommerfeld enhancement and coannihilations play important roles in the dark matter relic abundance calculation, reducing the upper bound on the large multiplet’s mass by almost a factor of 2. We determine the allowed parameter ranges including both of these effects and show that these models are as yet unconstrained by dark matter direct detection experiments, but will be probed by currently running and proposed future experiments. We also show that a Landau pole appears in these models at energy scales below $10^9\mathrm{GeV}$, indicating the presence of additional new physics below that scale.

Figure 1

The DM fraction ${\mathrm{\Omega }}_{\zeta }/{\mathrm{\Omega }}_{\mathrm{DM}}$ as a function of $m_{{\zeta }^{0,r}}$ for $n=6$ (left) and $n=8$ (right), computed solely from ${\zeta }^{0,r}{\zeta }^{0,r}\to \mathrm{SM}$ SM. The allowed region for ${\lambda }_2={\lambda }_3=0$ is shaded purple. The dashed cyan lines show where ${\mathrm{\Omega }}_{\zeta }/{\mathrm{\Omega }}_{\mathrm{DM}}=1$ for ${\mathrm{\Lambda }}_n={\mathrm{\Lambda }}_n^{\mathrm{MAX}}$ when ${\lambda }_{2,3}\ne 0$. As noted in the text, for the remainder of this paper we will set ${\lambda }_{2,3}=0$ for simplicity, so that ${\lambda }_4=\frac{2}{n}(-1{)}^{\frac{n}{2}}{\mathrm{\Lambda }}_n$.

Figure 2

Parameter values for which ${\mathrm{\Omega }}_{\zeta }/{\mathrm{\Omega }}_{\mathrm{DM}}=1$ for coannihilation (black solid) and single-species annihilation (dashed grey), as a function of both $m_{{\zeta }^{0,r}}$ and ${\mathrm{\Lambda }}_n$. The region above the horizontal dotted red line is excluded by the unitarity bound, ${\mathrm{\Lambda }}_n\le {\mathrm{\Lambda }}_n^{\mathrm{MAX}}$. We set ${\lambda }_2={\lambda }_3=0$.

Figure 3

Sommerfeld enhancement due to the Hulthén potential for ${\mathrm{\Lambda }}_n=0.1$ (left plot) and ${\mathrm{\Lambda }}_n=\pi$ (right plot). The curves correspond to different values of $\beta =v_{\mathrm{rel}}/c$. The solid blue curve corresponds to $\beta ={10}^{-5}$, the dashed red to $\beta ={10}^{-2}$, and the dotted green to $\beta ={10}^{-1}$.

Figure 4

Thermally averaged Sommerfeld enhancement factor from the Hulthén potential as a function of $m_{{\zeta }^{0,r}}$ and ${\mathrm{\Lambda }}_n$. Contours are labeled with the value of $⟨{\mathcal{S}}_h⟩$ for $x=20$. Since $⟨{\mathcal{S}}_h⟩$ does not depend on the size of the multiplet, the contours in the $n=6$ (left plot) and $n=8$ (right plot) models are the same. Except for numerical instability as $m_{{\zeta }^{0,r}}\to 0$ (i.e., $m_{{\zeta }^{0,r}}\approx m_{W,Z,h}$, so that the effective long-range potential vanishes), the Sommerfeld factor does not vary with $m_{{\zeta }^{0,r}}$, but does increase with increasing ${\mathrm{\Lambda }}_n$. The region above the horizontal dotted red line is excluded by the unitarity bound, ${\mathrm{\Lambda }}_n\le {\mathrm{\Lambda }}_n^{\mathrm{MAX}}$.

Figure 5

Parameter values for which ${\mathrm{\Omega }}_{\zeta }/{\mathrm{\Omega }}_{\mathrm{DM}}=1$ for Sommerfeld enhancement from a Higgs-exchange Hulthén potential (black solid) and single-species annihilation without the Sommerfeld effect (dashed grey), as a function of both $m_{{\zeta }^{0,r}}$ and ${\mathrm{\Lambda }}_n$. The region above the horizontal dotted red line is excluded by the unitarity bound, ${\mathrm{\Lambda }}_n\le {\mathrm{\Lambda }}_n^{\mathrm{MAX}}$.

Figure 6

Parameter values for which ${\mathrm{\Omega }}_{\zeta }/{\mathrm{\Omega }}_{\mathrm{DM}}=1$ comparing coannihilation and Sommerfeld enhancement. The leftmost (dashed magenta) curve corresponds to setting ${\mathcal{S}}_{\tau \phi }=1$ in Eq. (45) and allows us to see the effect of setting $v=0$. The middle (dotted blue) curve corresponds to ${\mathrm{\Omega }}_{\zeta }/{\mathrm{\Omega }}_{\mathrm{DM}}=1$ for coannihilations as in Fig. 2. The rightmost (solid black) curve corresponds to the full expression in Eq. (46).

Figure 7

Parameter values for which ${\mathrm{\Omega }}_{\zeta }/{\mathrm{\Omega }}_{\mathrm{DM}}=1$, with $n=6$ on the left and $n=8$ on the right. The solid black line incorporates both coannihilations and Sommerfeld enhancement [our final results, Eq. (46)]. The dotted-dashed orange line shows Sommerfeld enhancement with only a Higgs potential [Eq. (37)]. The dotted blue line shows coannihilations [Eq. (19)]. The dashed grey line shows the case of single-species annihilation without Sommerfeld effects [Eq. (11)]. The region above the horizontal dotted red line is excluded by the unitarity bound, ${\mathrm{\Lambda }}_n\le {\mathrm{\Lambda }}_n^{\mathrm{MAX}}$.

Figure 8

Location of the (one-loop) Landau pole in the $n=6$ (left panel) and $n=8$ (right panel) large multiplet models. The solid black curve shows $m_{{\zeta }^{0,r}}$ (bottom axis) for which ${\mathrm{\Omega }}_{\zeta }/{\mathrm{\Omega }}_{\mathrm{DM}}=1$ as in Fig. 7, while the dashed blue curve shows the location of the Landau pole (top axis) for the corresponding parameter point. We set ${\lambda }_2={\lambda }_3=0$ at the $m_{{\zeta }^{0,r}}$ scale as usual.

Figure 9

Normalized difference between the Landau pole scale $\mu$ and the ${\zeta }^{0,r}$ mass scale ${\mu }_0$ as a function of $|{\lambda }_4|$. In these figures, we set the initial conditions as in Eq. (47). The upper blue curve corresponds to $n=6$ and the lower red curve to $n=8$. The end point of each curve occurs at the unitarity bound on $|{\lambda }_4|$ from Table 1.

Figure 10

Direct detection predictions for the multi-TeV parameter space in the $n=6$ (blue) and $n=8$ (green) models. The solid lines show where ${\mathrm{\Omega }}_{\zeta }/{\mathrm{\Omega }}_{\mathrm{DM}}=1$ in each model, incorporating both Sommerfeld enhancement and coannihilations. The dashed lines on either side show the range of direct detection cross section for ${\mathrm{\Omega }}_{\zeta }/{\mathrm{\Omega }}_{\mathrm{DM}}\in [0.7,1.3]$ to account for the uncertainty in ${⟨\sigma \beta ⟩}_{\mathrm{STD}}$. The solid grey curve shows the extrapolation of the limit from the PandaX-II experiment [39], while the solid black curve shows the limit from LUX [40]. The dotted grey curves show extrapolations of the projected sensitivity of upcoming experiments, from top to bottom: DEAP-3600 [41], XENON1T [42], LZ [43], and DARWIN [44]. The shaded orange region at the bottom of the plot shows where the coherent scattering of neutrinos will become a non-negligible background, from Ref. [45].