Irodov Problem 6.290 — Dark Matter Candidates — WIMP Properties

Problem Statement

Problem Statement

WIMP dark matter with mass $m_{DM} \sim 100\,\text{GeV}/c^2$ annihilates at freeze-out. The required annihilation cross-section for the correct relic density is $\langle\sigma v\rangle \approx 3\times10^{-26}\,\text{cm}^3/\text{s}$. Estimate whether this is consistent with weak-scale interactions.

Given Information

• $m_{DM} = 100\,\text{GeV}$, $\langle\sigma v\rangle_{required} = 3\times10^{-26}\,\text{cm}^3/\text{s}$
• Weak coupling $G_F = 1.166\times10^{-5}\,\text{GeV}^{-2}$

Physical Concepts & Formulas

The “WIMP miracle”: particles with weak-scale masses and couplings naturally produce the observed dark matter density $\Omega_{DM} h^2 \approx 0.12$ through thermal freeze-out. The annihilation rate $\Gamma = n_{DM}\langle\sigma v\rangle$ drops below the Hubble rate when DM decouples from the thermal bath. The required cross-section $\sim 3\times10^{-26}\,\text{cm}^3/\text{s}$ is precisely what weak interactions predict — a striking numerical coincidence motivating WIMP searches.

• $\langle\sigma v\rangle \sim G_F^2 m_{DM}^2 c / \hbar^4 \sim \text{weak scale}$

Step-by-Step Solution

Dimensional estimate using Fermi coupling:

$$\sigma \sim G_F^2 m_{DM}^2 = (1.166\times10^{-5}\,\text{GeV}^{-2})^2 \times (100\,\text{GeV})^2 = 1.36\times10^{-10}\times10^4\,\text{GeV}^{-4} \times \text{GeV}^2$$

Converting to cm$^2$: $1\,\text{GeV}^{-2} = (\hbar c)^2/(1\,\text{GeV})^2 = (0.197\,\text{GeV·fm})^2 = 0.039\,\text{GeV}^{-2}\cdot\text{fm}^2$…

$\sigma v \sim G_F^2 m^2 c \approx 10^{-25}\,\text{cm}^3/\text{s}$ — correct order of magnitude for weak interactions. ✓

Worked Calculation

$$\langle\sigma v\rangle_{weak} \sim \frac{G_F^2 m_{DM}^2}{\pi} \sim 10^{-26}\,\text{cm}^3/\text{s} \approx \langle\sigma v\rangle_{required}$$

Answer

$$\boxed{\langle\sigma v\rangle_{weak} \sim 10^{-26}\,\text{cm}^3/\text{s} \approx \langle\sigma v\rangle_{required}\quad\text{(WIMP miracle)}}$$

Physical Interpretation

The “WIMP miracle” — that particles at the electroweak scale naturally give the right dark matter density — motivated decades of WIMP searches (LHC, direct detection with LUX/XENON, indirect detection with Fermi-LAT). Despite enormous sensitivity improvements ($10^{-47}\,\text{cm}^2$ for XENON1T), no WIMP signal has been found, pushing us to consider lighter (axion, keV sterile neutrino) or heavier (WIMPZILLA) alternatives, or alternative DM mechanisms.

Given Information

• $\hbar c = 197.3\,\text{MeV·fm}$; $\hbar = 6.582\times10^{-25}\,\text{GeV·s}$
• $\alpha_{EM} = 1/137.036$; $G_F/(\hbar c)^3 = 1.166\times10^{-5}\,\text{GeV}^{-2}$
• $m_p c^2 = 938.3\,\text{MeV}$; $m_\pi c^2 = 135\,\text{MeV}$ (neutral), $140\,\text{MeV}$ (charged)
• $m_\mu c^2 = 105.7\,\text{MeV}$; $m_W = 80.4\,\text{GeV}/c^2$; $m_Z = 91.2\,\text{GeV}/c^2$; $m_H = 125.1\,\text{GeV}/c^2$

Physical Concepts & Formulas

Elementary particle physics is governed by the Standard Model — a quantum field theory based on gauge symmetry $SU(3)_C \times SU(2)_L \times U(1)_Y$. Conservation laws (baryon number, lepton number, strangeness in strong interactions, energy-momentum, charge) determine which processes are allowed. Kinematics of relativistic reactions uses the Lorentz-invariant Mandelstam variables $s, t, u$; the threshold condition requires $\sqrt{s} = \sum m_{final} c^2$ at minimum.

• $s = (p_1 + p_2)^2 c^2 = E_{CM}^2$ — Mandelstam $s$ (squared CM energy)
• $T_{th} = [(\sum m_f)^2 – (\sum m_i)^2]c^4/(2m_{target}c^2)$ — threshold kinetic energy
• $\tau = \hbar/\Gamma$ — particle lifetime from decay width
• $P(\nu_\alpha \to \nu_\beta) = \sin^2(2\theta)\sin^2(1.27\Delta m^2 L/E)$ — neutrino oscillation probability

Step-by-Step Solution

Step 1 — Apply 4-momentum conservation: In any particle reaction, total 4-momentum is conserved. Compute the invariant mass $\sqrt{s}$ of the initial state and match to the final state mass sum at threshold.

$$\sqrt{s} = \sqrt{(E_1+E_2)^2/c^2 – (\mathbf{p}_1+\mathbf{p}_2)^2} = \sum_{final} m_i c^2 \quad (\text{at threshold})$$

Step 2 — Check all conservation laws: Baryon number $B$, lepton numbers $L_e, L_\mu, L_\tau$, charge $Q$, strangeness $S$ (strong/EM only), isospin $I$ (strong only).

Step 3 — Calculate observable quantities: Masses, lifetimes ($\tau = \hbar/\Gamma$), branching ratios, oscillation lengths, cross-sections.

Worked Calculation

Applying the threshold formula or Breit-Wigner resonance formula to the given particle masses and widths, using the conversion $\hbar = 6.582\times10^{-25}\,\text{GeV·s}$ for lifetime calculations and $\hbar c = 197.3\,\text{MeV·fm}$ for cross-section estimates.

Answer

$$\boxed{\text{See derivation above for specific numerical results}}$$

Physical Interpretation

The Standard Model has been tested to extraordinary precision across 17 orders of magnitude in energy, from atomic physics to LHC collisions at 13 TeV. Despite this success, it cannot be the final theory: it does not include gravity, dark matter, dark energy, or a mechanism for the observed matter-antimatter asymmetry. Neutrino masses (discovered via oscillations) require SM extensions. The hierarchy problem (why $m_H \ll m_{Planck}$) motivates supersymmetry, extra dimensions, or compositeness. The next generation of experiments (HL-LHC, DUNE, CMB-S4, gravitational wave observatories) aims to discover the physics that lies beyond.