Irodov Problem 6.285 — Particle Interactions — Feynman Diagrams

Problem Statement

Problem Statement

Draw (describe) the lowest-order Feynman diagrams for: (a) electron-muon scattering $e^- + \mu^- \to e^- + \mu^-$ (EM), (b) neutron beta decay $n \to p + e^- + \bar\nu_e$ (weak), (c) quark-quark scattering via gluon exchange (strong).

Given Information

• Mediators: $\gamma$ (EM), $W^\pm, Z^0$ (weak), $g$ gluons (strong)
• Coupling constants: $\alpha_{EM} = 1/137$, $\alpha_W \approx 1/29$, $\alpha_S \approx 0.1$–$1$ (running)

Physical Concepts & Formulas

The Standard Model has three gauge forces: electromagnetic (photon exchange, $\alpha = e^2/(4\pi\varepsilon_0\hbar c) = 1/137$), weak ($W^\pm, Z$ exchange, unification with EM at $E \gtrsim 100\,\text{GeV}$), and strong (8 gluons, SU(3) color gauge). Feynman diagrams provide a systematic perturbation theory where each vertex contributes a factor $\sqrt{\alpha}$.

• QED amplitude $\propto e^2 = 4\pi\alpha\hbar c$; propagator $\propto 1/(q^2-m^2c^2)$
• Fermi’s weak coupling: $G_F/(\hbar c)^3 = 1.166\times10^{-5}\,\text{GeV}^{-2}$

Step-by-Step Solution

(a) $e$-$\mu$ scattering: Single photon exchange ($t$-channel). The electron emits a virtual photon (4-momentum $q^\mu$); the muon absorbs it. Amplitude $\propto \alpha/q^2$.

(b) Neutron beta decay: The $d$ quark emits a virtual $W^-$; $d \to u + W^-$; then $W^- \to e^- + \bar\nu_e$. At energies $\ll m_W c^2 = 80.4\,\text{GeV}$, the $W$ propagator reduces to $G_F/\sqrt{2}$ (Fermi contact interaction).

(c) Quark-quark scattering: Gluon exchange in $t$-channel. The strong coupling $\alpha_S(Q^2)$ runs from $\sim 0.3$ at $Q = 1\,\text{GeV}$ to $\sim 0.1$ at $Q = 100\,\text{GeV}$ (asymptotic freedom).

Worked Calculation

Each diagram’s amplitude is given by the Feynman rules: fermion propagators $\propto 1/(\slashed p – m)$, boson propagators $\propto g_{\mu\nu}/(q^2 – M^2)$, vertices $\propto g_i \gamma^\mu$ (vector current coupling).

Answer

$$\boxed{\text{See description above: (a) }\gamma\text{-exchange, (b) }W^-\text{ exchange, (c) gluon exchange}}$$

Physical Interpretation

The Standard Model — quantum field theory of EM, weak, and strong interactions — is the most precisely tested theory in physics. QED predictions match experiment to 12 significant figures (electron $g-2$). The weak force unifies with EM at ~100 GeV (electroweak unification, Glashow-Weinberg-Salam, Nobel 1979). QCD’s asymptotic freedom (Nobel 2004) explains why quarks behave as free particles in high-energy collisions.

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.