Irodov Problem 6.281 — Conservation Laws — Particle Reactions

Problem Statement

Problem Statement

Determine which of the following reactions are forbidden and state which conservation law is violated: (a) $p \to e^+ + \gamma$, (b) $n \to p + e^- + \bar\nu_e$, (c) $\Lambda^0 \to p + \pi^-$, (d) $\Sigma^+ \to n + \pi^+$.

Given Information

• Baryon numbers: $B(p,n,\Lambda,\Sigma) = +1$; $B(\pi, e, \nu) = 0$; $B(\bar{p}) = -1$
• Strangeness: $S(\Lambda) = S(\Sigma) = -1$; $S(p,n,\pi) = 0$
• Lepton numbers: $L_e(e^-) = +1$, $L_e(\bar\nu_e) = -1$
• Electric charge: $Q(p) = +1$, $Q(n) = 0$, $Q(e^-) = -1$, etc.

Physical Concepts & Formulas

Conservation laws in particle physics: (1) Energy-momentum, (2) Electric charge, (3) Baryon number, (4) Lepton number (electron, muon, tau separately), (5) Strangeness (in strong/EM interactions; violated in weak). Any process violating a conservation law is forbidden.

• $\Delta B = 0$, $\Delta Q = 0$, $\Delta L_e = 0$ — always conserved
• $\Delta S = 0$ — conserved in strong/EM; $|\Delta S| = 1$ in weak

Step-by-Step Solution

(a) $p \to e^+ + \gamma$: $B: 1 \to 0$ (violated!). Also $L_e: 0 \to -1$ (violated). This proton decay is forbidden by baryon number conservation. (Experimentally: $\tau_p > 10^{34}$ yr.)

(b) $n \to p + e^- + \bar\nu_e$: $B: 1 \to 1+0+0$ ✓; $Q: 0 \to 1-1+0 = 0$ ✓; $L_e: 0 \to 0+1-1 = 0$ ✓; $\Delta S = 0$ ✓. Allowed (free neutron beta decay).

(c) $\Lambda^0 \to p + \pi^-$: $B: 1 \to 1$ ✓; $Q: 0 \to 1-1 = 0$ ✓; $S: -1 \to 0+0 = 0$ (violated by weak interaction — allowed). Allowed (weak decay, $\tau \sim 10^{-10}$ s).

(d) $\Sigma^+ \to n + \pi^+$: $B: 1 \to 1$ ✓; $Q: +1 \to 0+1 = +1$ ✓; $S: -1 \to 0+0 = 0$ (weak decay). Allowed.

Worked Calculation

Only (a) is forbidden — proton decay violates baryon number conservation. The others are all allowed (b by weak interaction, c and d by weak with $|\Delta S|=1$).

Answer

$$\boxed{\text{(a) Forbidden: violates baryon number and lepton number; (b),(c),(d) all allowed}}$$

Physical Interpretation

Baryon number conservation has been experimentally tested to extraordinary precision — the proton lifetime exceeds $10^{34}$ years. Grand Unified Theories (GUTs) predict proton decay via processes like $p \to e^+ + \pi^0$ at accessible rates — its non-observation rules out many GUT models. Super-Kamiokande has set the limit $\tau/B(p\to e^+\pi^0) > 1.6\times10^{34}$ yr.

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.