Irodov Problem 6.289 — Higgs Boson — Mass Generation

Problem Statement

Problem Statement

The Higgs boson was discovered at the LHC in 2012 with mass $m_H = 125.1\,\text{GeV}/c^2$. Estimate its decay width $\Gamma_H \approx 4\,\text{MeV}$ and lifetime. Identify its main decay modes.

Given Information

• $m_H = 125.1\,\text{GeV}/c^2$, $\Gamma_H = 4.07\,\text{MeV}$

Physical Concepts & Formulas

The Higgs boson is the quantum of the Higgs field — the mechanism by which $W^\pm$ and $Z$ bosons, and fermions, acquire mass. It couples to particles proportional to their mass: $g_{Hff} = m_f/v$ where $v = 246\,\text{GeV}$ is the Higgs vacuum expectation value. Main decays: $H \to b\bar b$ (58%), $H \to WW^*$ (21%), $H \to gg$ (9%), $H \to \tau^+\tau^-$ (6%), $H \to ZZ^*$ (3%), $H \to \gamma\gamma$ (0.2%).

• $\tau_H = \hbar/\Gamma_H$

Step-by-Step Solution

$$\tau_H = \frac{\hbar}{\Gamma_H} = \frac{6.582\times10^{-25}\,\text{GeV·s}}{4.07\times10^{-3}\,\text{GeV}} = 1.617\times10^{-22}\,\text{s}$$

Worked Calculation

$$\tau_H = \frac{\hbar}{\Gamma_H} \approx \frac{6.6\times10^{-25}}{4\times10^{-3}}\,\text{s} \approx 1.6\times10^{-22}\,\text{s}$$

Answer

$$\boxed{\tau_H \approx 1.6\times10^{-22}\,\text{s};\quad \text{main decay: }H\to b\bar b\,(58\%),\;H\to WW^*\,(21\%)}$$

Physical Interpretation

The Higgs lifetime of $1.6\times10^{-22}$ s is far too short to detect directly — only its decay products are observed. The dominant $H \to b\bar b$ mode is overwhelmed by QCD backgrounds at the LHC; instead, the discovery was made via $H \to \gamma\gamma$ and $H \to ZZ^* \to 4\ell$ (clean but rare modes). The Higgs discovery completed the Standard Model particle content and won the 2013 Nobel Prize (Higgs, Englert).

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.