Dark Matter VI: Beyond Standard Model

To explain the microscopic nature of dark matter requires physics beyond standard model. In fact, virtually all particle physicists believe that standard model is only an effective theory of some more fundamental theory. The following problems of standard model are the main motivation.

• Gauge hierarchy problem. Gauge hierarchy problem asked why Higgs mass ($\sim$100 GeV) is much smaller than Planck scale energy, the believed grand unification scale. A possible explanation is there is a TeV scale physics, which affects Higgs mass through quantum fluctuation.
  
  Moreover, calculation shows if Higgs mass is smaller than 134 GeV, the quantum fluctuation will cause the collapse of Higgs field vacuum. The current preliminary Higgs mass from LHC (125.7 GeV) demands a mechanism to stabilize the Higgs field vacuum. All these problem can be solved by TeV physics including super-symmetric models and universal extra dimension models.


• Neutrino mass, left-handed neutrino problem. In standard model, fermions acquire mass via Higgs mechanism. But fermion masses extend 14 orders of magnitude. This fact can hardly be viewed as a coincidence. Moreover, if neutrinos are massive, right-handed neutrinos must exist. One possibility is that there exist massive, non-interacting right-handed neutrinos, called sterile neutrinos. Sterile neutrino can solve the neutrino mass problem through see-saw mechanism.
  
  

  

  Fig. 1: fermion masses.



• Strong CP problem. In QCD, a term proportional to $\theta \epsilon_{\mu\nu\rho\lambda} F^{\mu\nu} F^{\rho\lambda}$ is allowed, which, if exists, will break CP-symmetry and lead to phenomena like nonzero neutron electric dipole moment. The current experimental observation constrains this term to be close to zero. Physicists do not believe this is a coincidence. Introducing axions can solve this problem

There have been many well motivated extension of standard model. Each of them is designed to solve one or more problems listed above.

Supersymmetry

Supersymmetric standard models introduce an additional symmetry between fermions and bosons called supersymmetry. Supersymmetry assigns each boson a fermionic supersymmetric partner. The lightest supersymmetric neutral particles (LSPs) are natural dark matter candidates, as protected by $R$-parity from decaying into lighter SM particles. If supersymmetry is broken at TeV scale, gauge hierarchy and Higgs vacuum stability problem can be solved elegantly.

Another strong support of supersymmetry comes from attempt towards the grand unification. Physicists postulate all interactions have the same origin at Plank scale $10^{18} \sim 10^{19}$ GeV. If this is true, strong weak and electromagnetism forces should be unified at around $10^{16}$ GeV. However, study of the evolution of three standard model couplings shows a discrepancy from unification at Planck scale. The discrepancy can be greatly reduced by introducing supersummetry (See Fig. 2).

Fig. 2: Left, evolution of couplings within standard model; Right, evolution of couplings by introducing a pair of vector like fermions carrying SM charges and masses of order 300 GeV-1 TeV.

For all these reasons, supersymmetric models are among the most favored standard model extensions. But one common issue of these models is that they have a vast set of parameters. The Minimal Supersymmetric Standard Model (MSSM) for example already has 63 parameters. It is impossible to explore the whole parameter space. By using theoretical arguments and/or current experimental data, majority of the parameter can be fixed to certain reasonable values. The left parameter set, containing 5 - 10 parameters, can be constrained by experiment from the upcoming particle physics experiments and astrophysics observation if the model is designed for dark matter (See Fig 3).

Fig. 3: The parameter space of various MSSM models. The shaded area is excluded by measurement of $B_s \to \mu^+ + \mu^-$ and $B_d → \mu^+ \mu^-$ branching ratios.

Extra Dimensions

Extra Dimension models (ED) assume there exist extra dimensions besides the usual 3+1 spacetime. The shape of the extra dimensions are usually curled, such as a small circle, to explain their invisibility. In ED, each normal particle corresponds to a set of excitations in the extra dimension, known as Kaluza-Klein (KK) tower. The mass of KK excitations in the tower, \[ m^2_\mathrm{KK} = m^2_\mathrm{SM} + n^2 / R^2, \quad n = 0, 1, 2, \dots \] where $R$ is the radius of the extra dimension, $R^{-1} \gtrsim$ 300 GeV to explain why they have not been observed in current colliders. Lightest KK particles (LKPs) are also natural dark matter candidates, as protected by KK-parity. Meanwhile, if the radius of the extra dimension is about $\mathrm{TeV}^{-1}$ , ED extension of standard model known as Universal Extra Dimension (UED) can replace Higgs mechanism to break the electroweak gauge symmetry, hence solves the gauge hierarchy problem and Higgs vacuum stability problem.

Just like supersymmetry, ED may also modify gravity. In the large extra dimension model of Arkani-Hamed, Dimopoulos and Dvali (ADD), standard model is lived on TeV scale 4D surface (called brane), whereas gravity and only gravity can penetrate from Planck scale brane and the TeV scale brane by propa- gating in the extra dimensions. EDs are constrained by various collider experiments, gravity tests and dark matter relic density. In theminimal case (mUED), UED only has one free parameter, the radius R of the curled extra dimension (See Fig. 4).

Fig. 4: Combined collider constraints on mUED. MH is standard model Higgs mass. By Belanger et al.(2012)

Peccei-Quinn Theory

Peccei-Quinn theory is proposed to solve the strong CP problem in QCD. Peccei- Quinn theory postulates the coupling constant θ of strong CP term is a dynamical field with a new global symmetry Peiccei-Quinn symmetry. The new field is called axion. In some models, axion is also a dark matter candidate.

See-Saw Mechanism and Neutrino Masses

To explain neutrino mass, one may extend standard model to include neutrino mass via Higgs mechanism as other fermions. So one need a right-handed neutrino. Right-handed neutrino is a isospin singlet. It does not interaction with gauge bosons. Meanwhile, it can also acquire a Majorana mass term. \[ \mathcal{L} = m_D \bar{\nu}_R \nu_L + m_M \bar{\nu}^c_L \nu_R + \mathrm{h.c.} \] The mass matrix is, \begin{equation} \left( \begin{array}{c c} 0 & m_D \\ m_D & m_M \\ \end{array} \right) \end{equation} $m_M \sim M_\text{Pl} \gg m_D$. there are two mass eigenstates, $m_1 \simeq m_M, m_2 \simeq -\frac{m_D^2}{m_M}$. $m_2 \ll m_D \ll m_M$. This is the so-called see-saw mechanism. It explains why neutrino mass $m_2$ is much smaller than fermion mass $m_D$. The first mass eigenstate is a massive nearly non-interacting neutrino, called sterile neutrino. Sterile neutrino is a very good dark matter candidate.

It should be noted that these models may appear together, even producing new dark matter candidates. For example, the supersymmetric partner of axion, called axino is also a dark matter candidate.

Weakly Interacting Massive Particles

A large class of dark matter candidates is weakly interaction massive particles (WIMPs). WIMPs are neutral stable particles with mass of weak scale $\sim$ 100 GeV, and coupling with standard model particles on weak coupling scale $\alpha \sim 10^{-2}$. Candidates in WIMP class include lightest supersymmetric particles (LSPs) such as neutralinos (the linear superposition of photino, Zino and Higgsino), lightest Kaluza-Klein excitations (LKPs) from universal extra dimension models.

WIMPs may be produced in early universe as the thermal relic of Big Bang. At very early universe, WIMPs are in equilibrium. As the temperature falls below its $m_{\chi}$ , there are more WIMPs annihilating into lighter particles than the converse reaction. The WIMP density drops exponentially, until the reaction rate below the Universe expansion. Then WIMPs freeze-out from equilibrium as the thermal relic and their co-moving number density remains fixed as the relic density. The above description can be formally modeled by Boltzmann equation: \[ \frac{\mathrm{d}n_\chi}{\mathrm{d} t} + 3 H(t) n_\chi = -\left< \sigma_A v \right> (n_\chi^2 - \bar{n}^2_\chi ) \] where $H$ is the Hubble parameter, $\bar n$ is the equilibrium number density.  The equation can be solved numerically. In the simplest case, the frozen-out temperature is about $T_f \sim m_\chi^2 /20$ and the relic density is, \[
\Omega_\chi h^2 \simeq \frac{ 3\times 10^{-27} \mathrm{cm^3 s^{-1}}}{\left< \sigma_A v \right>}
\]. The cross section of WIMPs is about $\left< \sigma_A v \right> \sim \frac{\alpha^2}{m_\chi^2} (a + b v^2 + \mathcal{O}(v^4)) $. The first two terms represent $s$-wave and $p$-wave cross section respectively. $v \ll 1$, so higher order terms can be neglected. For the case of s-wave dominated annihilation, $a \sim \mathcal{O}(1)$, then $\Omega_\chi \sim 10^{-3} - 10^{-1}$, where we have allowed one order of magnitude fluctuation. The cursory estimation shows,  particles with weak scale mass and coupling have naturally the correct relic density as dark matter. This is called the "WIMP miracle". These properties make WIMPs the leading dark matter candidates.

Fig. 4: The WIMP relic density. The dashed line is relic density without thermal frozen-out. The solid lines are the actual relic density. The strips represents a variation of the cross section of orders of magnitude from the one with the "correct relic density".