This section summarizes the theoretical and Higgs phenomenological constraints on the scalar sector parameters space of G2HDM model discussed in [26].
This study focused on constraining the parameter space of the potential under several physical arguments and phenomenological results. The first physical consideration is the boundedness of the scalar potential. This relies on the fact that the potential must be bounded from below to ensure the vacuum stability (VS) of the theory. Second, one needs to examine whether the theory satisfies the perturbative unitarity (PU) or not. This kind of check can be realized by calculating the scattering amplitudes at very high energy. If the amplitudes are well behaved and do not go to infinity as the energy gets higher, then the unitarity of the theory is maintained. In the scalar sector of G2HDM model, this can be achieved by considering all possible 2→ 2 scattering amplitudes in the scalar sector and evaluate them in the high energy limit. Thus, all scalars appear in this model need to be taken into account for these scattering amplitudes. Finally, the experimental constrains coming from the LHC Higgs data are needed to further limit the allowed parameter space of the model.
In [26], the surviving parameter space of the model have been evaluated under the theoretical constraints from vacuum stability (VS), perturbative unitarity (PU) and the 125 GeV Higgs physics (HP) data including the Higgs boson mass and signal strengths of Higgs boson decays into diphoton and τ+τ− from the LHC. The study showed that out of the eight λ−parameters, only two of them λH and λHΦare essentially constrained by (VS+PU+HP).
Other couplings like λH′, λHΦ′ and λΦ∆ are less constrained. This study also concluded that some of the parameters such as MH∆, MΦ∆ and the VEVs are constrained only by HP but not by (VS+PU). In the numerical set up for the scanning in [26], the two parameters MH∆, MΦ∆
are varied in the range of [−1,1] TeV, v∆∈ [0.5,20] TeV, while v and vΦwere fixed at 246 GeV and 10 TeV respectively.
10−4
Fig. 3.1 A summary of the parameter space allowed by the theoretical and phenomenological constraints. The red regions show the results from the theoretical constraints (VS+PU).
The magenta regions are constrained by Higgs physics as well as the theoretical constraints (HP+VS+PU). Figure is taken from [26].
We show a summary of allowed regions of parameter space in Fig. 3.1. The upper red triangular block corresponds to (VS+PU) constraints, while the lower magenta triangular block corresponds to the (VS+PU+HP) constraints. The diagonal panels indicate the allowed ranges of the eight couplings λH,Φ,∆, λH′, and λHΦ,H∆,Φ∆, λHΦ′ under the combined constraints of (VS+PU+HP).
Chapter 4
WIMP Dark Matter and Its Constraints
The summary of the particle dark matter along with their experimental constraints will be discussed in this chapter. We start with the astrophysical evidence of the dark matter which provides a strong motivation in studying this open problem. Next, we try to explain the dark mater problem from particle physics point of view by studying its thermal evolution in the universe. Finally, the corresponding experimental constraints coming from the dark matter searches in the lab and sky will be given.
4.1 Astrophysical Evidence of the Dark Matter
There are many evidences that support the existence of the dark matter. Most of the evidences come from astrophysical observations such as galactic rotation curves, gravitational lensing, structure formation, the observed cosmic microwave background (CMB), baryon acoustic oscillation and the matter power spectrum, etc. In this section, we will discus the galactic rotation curves and gravitational lensing which provide strong evidence of the dark matter.
4.1.1 Galactic Rotation Curves
One of the strongest pieces of evidence for DM comes from studying the rotational velocity of stars which measure the circular speed of the star orbiting a particular galaxy. It is known that stars orbiting the galaxy is dictated by the gravitational potential. From Newtonian gravity, using the fact that the gravitational force is balanced by centrifugal force of the star, one can extract the star’s circular velocity, vc, as
vc(r) =
rGM
r (4.1)
1980ApJ...238..471R
Fig. 4.1 Rotation curves of spiral galaxies as observed by Rubin et al. [31]. At large radial distance from the center, most of galaxies exhibit constant circular velocity.
where M is the mass enclosed by the galaxy, r is the distance between galaxy and the star, and G is the Newton’s gravitational constant. If the star located outside the galactic disk, M will be constant as there is no other massive object around. In this case vc∝ r−1/2. However, the results of observations concluded that the circular velocity curve is constant or M(r) ∝ r.
This gives us a hint that there is unknown matter apart from the visible matter and this unseen matter distributes outside the galaxy disk. Figure 4.1 displays the 21 Sc rotation curves as observed by Rubin et al. in [31]. This clearly shows that at large distance from the galactic center, the circular velocities are approaching constant values.
According to the observed rotation curves, we can conclude that the dark matter mass density distribution is given by
ρ (r) ∝ M(r) r3 ∼ 1
r2. (4.2)
This conclusion was made by assuming that the dark matter is spherically symmetric dis-tributed around the galaxy center in contrast to the visible matter that mostly located at the galactic disk.
4.1.2 Gravitational Lensing
It is well known from Einstein’s theory of General Relativity that light will be bent when it propagates under the influence of gravitational potential. Thus, one expects the trajectory of the light will be deviated around massive objects. This is the main idea of the gravitational
lensing. The deflection angle δ φ of the light around an object with the mass M is given by
δ φ ≈4GNM
b , (4.3)
where GN is the usual Newton’s constant and b is the impact parameter. It is clear that when the mass M equals to zero (no massive object), light will follow straight line during its propagation. This equation also tells us that when the deflected angle is known (to be exact observed), we can further obtain the information about the mass of the corresponding object.
The incidental light from a source will pass through the massive object before it gets to the observer’s "eyes". When the observer, the massive object as well as the light source are aligned in a straight line, the light is focused as if it passed through optical lenses. Therefore, the observer will see multiple images which originated from the same object and he/she will be able to obtain its mass distribution. The observations of the galaxies in [32] exhibit the density profile ρ(r)≈ r12 consistent with the one extracted from galactic rotation curves.