Renormalization group equations of a cold dark matter two-singlet model

We study via the renormalization group equations at one-loop order the perturbativity and vacuum stability of a two-singlet model of cold dark matter (DM) that consists in extending the Standard Model with two real gauge-singlet scalar fields. We then investigate the regions in the parameter space in which the model is viable. For this, we require the model to reproduce the observed DM relic density abundance, to comply with the measured XENON 100 direct-detection upper bounds, and to be consistent with the renormalization group equation perturbativity and vacuum stability criteria up to 40 TeV. For a small mixing angle $\theta$ between the physical Higgs $h$ and auxiliary field, and DM-$h$ mutual coupling constant ${\lambda }_0^{(4)}$, we find that the auxiliary field mass is confined to the interval 116–138 GeV, while the DM mass is mainly confined to the region above 57 GeV. Increasing $\theta$ enriches the existing viability regions without relocating them, while increasing ${\lambda }_0^{(4)}$ shrinks them with a tiny relocation. We show that the model is consistent with the recent Higgs-boson-like discovery by the ATLAS and CMS experiments, while very light dark matter (masses below 5 GeV) is ruled out by the same experiments.

Figure 1

The running of the self-couplings (scalars only). The self-coupling ${\eta }_1$ of the auxiliary field ${\chi }_1$ dominates over the Higgs self-coupling $\lambda$.

Figure 2

Running of the mutual couplings (scalars only). For these values of the parameters, all three are well below the self-couplings.

Figure 3

Running of the self-couplings (scalars only) for a larger DM mass. ${\eta }_1$ is still dominant, even if ${\eta }_0$ picks up faster behind it.

Figure 4

Running of the mutual couplings (scalars only) for a larger DM mass. ${\eta }_{01}$ starts above 1, much higher than the two others, even higher than the self-coupling ${\eta }_1$.

Figure 5

Running of the self-couplings (scalars only) with a larger physical Higgs self-coupling ${\lambda }_0^{(4)}$. The self-coupling ${\eta }_1$ is still dominant.

Figure 6

Running of mutual couplings (scalars only). Larger ${\lambda }_0^{(4)}$ helps ${\eta }_{01}$ rise well above ${\lambda }_0$ and ${\lambda }_1$, but not enough to win over the self-coupling ${\eta }_1$.

Figure 7

Running of the self-couplings (full RGE). ${\eta }_1$ controls perturbativity, and the Higgs coupling $\lambda$ becomes negative quickly.

Figure 8

Running of the Higgs self-coupling $\lambda$ (full RGE). It gets negative at about 15 TeV for this set of parameters’ values.

Figure 9

Running of the mutual couplings (full RGE). The inclusion of the other SM particles flattens the runnings.

Figure 10

Running of the self-couplings (full RGE) with $m_0$ larger. ${\eta }_1$ is more closely tailed by ${\eta }_0$, and $\lambda$ decreases and turns negative at about 10 TeV.

Figure 11

Running of the mutual couplings (full RGE) with $m_0$ larger. ${\eta }_{01}$ starts well above ${\lambda }_0$ and ${\lambda }_1$ and leaves the perturbativity region before the self-coupling ${\eta }_1$.

Figure 12

Regions of viability of the two-singlet model (in blue). Physical Higgs self-coupling ${\lambda }_0^{(4)}$ and mixing angle $\theta$ are very small.

Figure 13

Regions of viability (blue) of the model. ${\lambda }_0^{(4)}$ is still very small, but $\theta$ is larger. The region is richer, but not relocated.

Figure 14

The region of viability (blue) is even richer for a larger mixing angle $\theta$.

Figure 15

The physical Higgs self-coupling ${\lambda }_0^{(4)}$ shrinks the viability region (blue) as it increases.

Figure 16

Viability regions when implementing the three constraints successively. Cyan: excluded by the relic density constraint. Cyan and green: excluded by both the relic density and direct-detection constraints. Black: what stays viable when adding the RGEs, the same region as in Fig. 14.

Figure 17

Behavior of the RGE control parameters ${\eta }_1$, ${\eta }_{01}$, and $\lambda$ at ${\Lambda }_m=40\mathrm{GeV}$ as functions of $m_1$. The Higgs self-coupling enters positive values at about $m_1\simeq 116\mathrm{GeV}$ and ${\eta }_1$ leaves perturbativity at about $m_1\simeq 138\mathrm{GeV}$. This trend is not very sensitive to large $m_0$.