Complex scalar singlet dark matter: Vacuum stability and phenomenology

We analyze one-loop vacuum stability, perturbativity, and phenomenological constraints on a complex singlet extension of the standard model scalar sector containing a scalar dark matter candidate. We study vacuum stability considerations using a gauge-invariant approach and compare with the conventional gauge-dependent procedure. We show that, if new physics exists at the TeV scale, the vacuum stability analysis and experimental constraints from the dark matter sector, electroweak precision data, and LEP allow both a Higgs-like scalar in the mass range allowed by the latest results from CMS and ATLAS and a lighter singlet-like scalar with weak couplings to standard model particles. If instead no new physics appears until higher energy scales, there may be significant tension between the vacuum stability analysis and phenomenological constraints (in particular electroweak precision data) to the extent that the complex singlet extension with light Higgs and singlet masses would be ruled out. We comment on the possible implications of a scalar with $\sim 125\mathrm{GeV}$ mass and future ATLAS invisible decay searches.

Figure 1

A plot of ${\delta }_2^2(M_Z)$ vs $\lambda (M_Z)d_2(M_Z)$. For all points, $M_A=10\mathrm{GeV}$, $x=100\mathrm{GeV}$, and $d_2(M_Z)=0.2$. The tree level vacuum stability requirement, ${\delta }_2^2<\lambda d_2$, is indicated with the solid line. All points satisfy the effective potential vacuum stability requirement of Eq. (12) with $\Lambda =1\mathrm{TeV}$ for some choice of gauge parameter $\xi$. For gray points, $\xi =0$ (Landau gauge); for dark gray points, $\xi =1$; and for light gray points, $\xi =50$. [In this and subsequent figures, the point (0, 0) is included for reference only.]

Figure 2

Feynman diagrams showing processes contributing to the annihilation cross section of the dark matter particles, $A$.

Figure 3

Plots of ${\delta }_2^2(M_Z)$ vs $\lambda (M_Z)d_2(M_Z)$. For all plots, $M_A=10\mathrm{GeV}$, $x=100\mathrm{GeV}$, and $d_2(M_Z)=0.2$. The tree level vacuum stability requirement, ${\delta }_2^2<\lambda d_2$, is indicated with the solid line. Gray points satisfy the constraints on the running couplings, Eq. (18), while black points satisfy the effective potential vacuum stability requirement, Eq. (12), in the Landau gauge. In the left column we take ${\delta }_2>0$ while in the right column ${\delta }_2<0$. The cutoff scale $\Lambda$ is 1 TeV (top row), 1000 TeV (middle row), or ${10}^{15}\mathrm{GeV}$ (bottom row).

Figure 4

Results of the scan for $M_A=10\mathrm{GeV}$, $x=100\mathrm{GeV}$, $d_2(M_Z)=0.2$, and $\Lambda =1\mathrm{TeV}$ with ${\delta }_2<0$ shown in the $M_{S^{\prime }}$ vs $M_{h^{\prime }}$ plane. Dark colored points oversaturate the relic density, while light colored points (under)saturate. The top left plot imposes only the RG coupling limits, Eq. (18). RG coupling limits plus either the LEP constraints (top right) or the EWPO constraints (bottom left), and finally all three (bottom right), are also shown.

Figure 5

Same as Fig. 4 but with $\Lambda ={10}^6\mathrm{GeV}$.

Figure 6

Same as Fig. 4 but with $\Lambda ={10}^{15}\mathrm{GeV}$.

Figure 7

Results of the scan for $M_A=10\mathrm{GeV}$, $x=100\mathrm{GeV}$, $d_2(M_Z)=0.2$, and $\Lambda =1\mathrm{TeV}$ with ${\delta }_2<0$ shown in the $M_{S^{\prime }}$ vs $M_{h^{\prime }}$ plane. Dark colored points oversaturate the relic density, while light colored points (under)saturate. The top left plot imposes only the RG coupling limits, Eq. (18). RG coupling limits plus either the XENON100 bound (top right) or satisfaction of the requirement for ATLAS invisible Higgs decay searches (bottom left), and finally all three (bottom right), are also shown.

Figure 8

Same as Fig. 7 but with $\Lambda ={10}^6\mathrm{GeV}$.

Figure 9

Same as Fig. 7 but with $\Lambda ={10}^{15}\mathrm{GeV}$.

Figure 10

Here are shown the effects on the mass eigenvalues and the relic density of changing $d_2$ and $x$ (with ${\delta }_2<0$). Only the RG running constraints ($\Lambda =1\mathrm{TeV}$) have been imposed. Left column ($M_{S^{\prime }}<10\mathrm{GeV}$, red in color version), $x=10\mathrm{GeV}$; middle column ($10<M_{S^{\prime }}<100\mathrm{GeV}$, blue in color version), $x=100\mathrm{GeV}$; right column ($M_{S^{\prime }}>100\mathrm{GeV}$, green in color version), $x=1000\mathrm{GeV}$. Top row, $d_2=0.2$; middle row, $d_2=0.5$; bottom row, $d_2=0.9$. Dark colored points oversaturate the relic density, while light colored points (under)saturate.

Figure 11

Here are shown the allowed mass eigenvalues ($M_A=10\mathrm{GeV}$) after imposing the RG evolution requirement in Eq. (18) (central column) plus the LEP and EWPO constraints (right column) or, alternately, the direct detection bound from XENON100 and the invisible decay requirements (left column). The effect of increasing the cutoff scale is also seen: $\Lambda =1\mathrm{TeV}$ (top row), ${10}^6\mathrm{GeV}$ (middle row), and ${10}^{15}\mathrm{GeV}$ (bottom row). Note that the top central plot corresponds to overlaying all plots in Fig. 10. The value of $x$ determines $M_{S^{\prime }}$ as in Fig. 10 with $x=10\mathrm{GeV}$ for $M_{S^{\prime }}<10\mathrm{GeV}$, $x=100\mathrm{GeV}$ for $10<M_{S^{\prime }}<100\mathrm{GeV}$, and $x=1000\mathrm{GeV}$ for $M_{S^{\prime }}>100\mathrm{GeV}$. The appearance of discrete bands for a given $x$ is the result of the three different choices for $d_2$.

Figure 12

The same as Fg. 11 but with a dark matter mass of 100 GeV.