Heavy right-handed neutrinos and dark matter in the $\nu \mathrm{CMSSM}$

We perform a systematic study of the effects of the type-I seesaw mechanism on the dark matter abundance in the constrained minimal supersymmetric standard model (CMSSM) which includes three right-handed neutrinos (the $\nu \mathrm{CMSSM}$). For large values of $m_0$, $m_{1/2}$, we exploit the effects of large neutrino Yukawa couplings on the renormalization group evolution of the up-type Higgs. In particular, we show that the focus point scale can greatly exceed the electroweak scale resulting in the absence of a focus point region for which the relic density of neutralinos is within the range determined by WMAP. We also discuss the effects of the right-handed neutrinos on the so-called funnel region, where the relic density is controlled by $s$-channel annihilations through a heavy Higgs. For small values of $m_0$, $m_{1/2}$, we discuss the possibility of sneutrino coannihilation regions with an emphasis on the suppression of the left-handed slepton doublet masses due to the neutrino Yukawa coupling. We consider two types of toy models consistent with either the normal or inverted hierarchy of neutrino masses.

Figure 1

The RG evolutions of $m_{H_u}$ with different values of $m_0$ for the CMSSM (dashed) and the $\nu \mathrm{CMSSM}$ with $M_{N_3}={10}^{15}\mathrm{GeV}$ and $m_{{\nu }_3}=0.05\mathrm{eV}$ (solid). We have set $m_{1/2}=300\mathrm{GeV}$, $\tan \beta =10$, $A_0=0$, and $\mu >0$.

Figure 2

The $(m_0,M_{N_3})$ plane for fixed $m_{1/2}=300\mathrm{GeV}$, $\tan \beta =10$, $A_0=0$, and $m_{{\nu }_3}=0.05\mathrm{eV}$. The (pink) shaded region shows the portion of parameter space where the radiative electroweak symmetry breaking conditions cannot be satisfied. The focus point region (turquoise) compatible with Eq. (5) runs along this boundary nearly coinciding with the contour for $\mu =200\mathrm{GeV}$. Other contours of constant $\mu$ are also shown. The (red) dot-dashed curve shows the contour of $m_h=114.4\mathrm{GeV}$. Below this curve, the Higgs mass is below the LEP limit. To the right of the vertical dashed line, $y_{N_3}^2/16{\pi }^2>1/10$ and our perturbative expansion becomes suspect.

Figure 3

The $(m_0,m_{1/2})$ plane for $\tan \beta =10$, $A_0=0$, $m_{{\nu }_3}=0.05\mathrm{eV}$, and four choices of $M_{N_3}$. The region labeled $Q_{\mathrm{GUT}}$ corresponds to the CMSSM. The black solid lines represent the boundaries of the regions above which (i.e. for bigger $m_0$) there is no electroweak symmetry breaking and ${\mu }^2<0$. Contours and shaded regions are described in the text.

Figure 4

The $(m_0,M_{N_3})$ plane for fixed $m_{1/2}=1100\mathrm{GeV}$, $\tan \beta =55$, $A_0=0$, and $m_{{\nu }_3}=0.05\mathrm{eV}$. The funnel region (turquoise) compatible with Eq. (5) runs almost horizontally around $m_0\simeq 1100\mathrm{GeV}$. Contours of constant $m_A/2m_{\chi }$ are also shown. To the right of the vertical dashed line, $y_{N_3}^2/16{\pi }^2>1/10$ and our perturbative expansion becomes suspect.

Figure 5

As in Fig. 3, the funnel regions for the CMSSM and for different values of $M_{N_3}=2\times {10}^{13}$, $5\times {10}^{13}$, ${10}^{14}\mathrm{GeV}$ in the $\nu \mathrm{CMSSM}$.

Figure 6

As in Fig. 2, the $(m_0,M_{N_1})$ plane for fixed $m_{1/2}=300\mathrm{GeV}$, $\tan \beta =10$, $A_0=0$, and $m_{{\nu }_1}=m_{{\nu }_2}=0.05\mathrm{eV}$. It is assumed that $M_{N_1}=M_{N_2}$.

Figure 7

The $(m_0,m_{1/2})$ plane in the $\nu \mathrm{CMSSM}$. In both panels, $\tan \beta =10$ and $A_0=1000\mathrm{GeV}$. In the left panel, $M_{N_3}={10}^{15}\mathrm{GeV}$ and corresponds to the case of a normal neutrino mass hierarchy while in the right panel, $M_{N_1}=M_{N_2}={10}^{15}\mathrm{GeV}$, corresponding to an inverted hierarchy. Contours and shading are described in the text.