Precision Tests of the Standard Model

13.1 The $g$ -2 Anomaly

The anomalous magnetic moment of the electron and muon:

a_e = \frac{g_e - 2}{2}, \quad a_\mu = \frac{g_\mu - 2}{2}

The Dirac equation predicts $g = 2$ exactly, but QED radiative corrections give:

a_e^{\text{QED} = \frac{\alpha}{2\pi} - 0.328\,478\,966\left(\frac{\alpha}{\pi}\right)^2 + 1.181\,241\,456\left(\frac{\alpha}{\pi}\right)^3 - 1.9144(35)\left(\frac{\alpha}{\pi}\right)^4}

The experimental value agrees with theory to 12 significant figures, making $a_e$ the most precisely verified prediction in all of physics.

The muon $g$ -2: The muon is $\sim 207$ times heavier than the electron, so it is more sensitive to virtual particles beyond the Standard Model (supersymmetry, dark photons, etc.).

a_\mu^{\text{exp} - a_\mu^{\text{SM} = (251 \pm 59) \times 10^{-11}}}

This $\sim 4.2\sigma$ discrepancy (as of 2023) is one of the strongest hints of physics beyond the SM.

13.2 Electroweak Precision Observables

The $Z$ -pole observables measured at LEP and SLC test the SM at the per-mil level:

$m_Z = 91.1876 \pm 0.0021$ GeV
$\Gamma_Z = 2.4952 \pm 0.0023$ GeV (total $Z$ width)
$\sin^2\theta_{\text{eff}^{\text{lept} = 0.23155 \pm 0.00016}}$ (effective weak mixing angle)
$R_\ell = \Gamma_{\text{had}/\Gamma_{\ell\ell} = 20.767 \pm 0.025}$ (hadronic to leptonic width ratio)
$A_{FB}^{0,\ell} = 0.0171 \pm 0.0010$ (forward-backward asymmetry)

The $S$ , $T$ , $U$ parameterisation (Peskin, Takeuchi) provides a model-independent framework for comparing these measurements:

\alpha_{\text{em}(m_Z) = \frac{\sqrt{2}G_F m_W^2(1 - m_W^2/m_Z^2)}{\pi\alpha} \times \frac{1}{1 - \Delta r}}

Where $\Delta r$ is the radiative correction depending on $S$ , $T$ , $U$ . Current data give $S = 0.05 \pm 0.11$ and $T = 0.09 \pm 0.13$ Consistent with the SM ( $S = T = 0$ ) but leaving room for new physics.

13.3 Rare Decays and Flavour Physics

$B$ -physics anomalies. The LHCb experiment has observed several tensions in $B$ -meson decays:

$R_{K^{(*)}}$ : The ratio $R_K = \text{BR}(B^+ \to K^+\mu^+\mu^-)/\text{BR}(B^+ \to K^+e^+e^-)$ is predicted to be 1 in the SM (lepton universality). Measurements show $R_K = 0.846^{+0.044}_{-0.041}$ ( $3.1\sigma$ deviation).
$b \to s\ell^+\ell^-$ angular observables: The observable $P_5"$ shows a persistent deviation from SM predictions.

These anomalies could indicate lepton-flavour-universal new physics (e.g., a $Z'$ boson coupling preferentially to muons).

Kaon physics: The extremely rare decay $K_L \to \mu^+\mu^-$ has been observed with BR $\sim 3 \times 10^{-11}$ (SM prediction), constraining new physics at the TeV scale through the process $s \to d\ell^+\ell^-$ .

13.4 Neutrinoless Double Beta Decay

The decay $0\nu\beta\beta$ : $(A, Z) \to (A, Z+2) + 2e^-$ violates lepton number by two units. If observed, it would prove that neutrinos are Majorana particles (identical to their antiparticles).

The half-life:

(T_{1/2}^{0\nu})^{-1} = G_{0\nu}|M_{0\nu}|^2\frac{\langle m_{\beta\beta}\rangle^2}{m_e^2}

Where $G_{0\nu}$ is the phase space factor, $M_{0\nu}$ is the nuclear matrix element, and $\langle m_{\beta\beta}\rangle$ is the effective Majorana mass.

Current best limit: $T_{1/2}^{0\nu} > 1.8 \times 10^{26}$ yr ( $^{76}$ Ge, GERDA), corresponding to $\langle m_{\beta\beta}\rangle < 0.07$ — $0.16$ eV.

Worked Example 13.1: QED Correction to Electron $g$-Factor

The leading QED correction to $a_e$ :

a_e^{(1)} = \frac{\alpha}{2\pi} = \frac{1/137.036}{2\pi} = 0.001161 \times 10^{-3}

The full QED + hadronic + weak correction:

a_e^{\text{total} = 1\,159\,652\,180.73(0.28) \times 10^{-12}}

Experimental (Gabrielse group, Harvard, 2023):

a_e^{\text{exp} = 1\,159\,652\,180.59(0.22) \times 10^{-12}}

The agreement is at the level of $0.2 \times 10^{-12}$ out of $1160 \times 10^{-9}$ I.e., relative precision of $1.7 \times 10^{-13}$ . This is the most precise test of any prediction in physics.

The comparison also determines $\alpha$ to higher precision than any direct measurement:

\alpha^{-1} = 137.035\,999\,166(15)

Worked Examples

Example 1: Conservation laws

Problem. Is the decay $\Lambda^0 \to p + \pi^-$ possible? Check all conservation laws.

Solution. Charge: $0 = +1 + (-1)$ ✓. Baryon number: $1 = 1 + 0$ ✓. Lepton number: $0 = 0 + 0$ ✓. Strangeness: $-1 \neq 0 + 0$ ✗ (violated, but strangeness is not conserved in weak decays). The decay is possible via the weak interaction.

$\blacksquare$

Example 2: Hubble’s law

Problem. A galaxy has redshift $z = 0.05$ . If $H_0 = 70 \mathrm{ km/s/Mpc}$ , estimate its distance.

Solution. $v \approx cz = 0.05 \times 3 \times 10^5 = 1.5 \times 10^4 \mathrm{ km/s}$ . $d = v/H_0 = 15000/70 = 214 \mathrm{ Mpc}$ .

$\blacksquare$

Intuition

Precision tests of the Standard Model are like checking a clock against atomic time. The electron’s magnetic moment agrees with QED to twelve decimal places, making it the most precise prediction in physics. The muon g-2 discrepancy is like finding a tiny wobble in a precision instrument, suggesting unseen particles contributing virtual effects. Electroweak precision observables test the self-consistency of the Standard Model’s gauge structure. Rare decays probe energy scales far beyond direct collider reach, like hearing echoes from distant thunder. Any deviation from Standard Model predictions would signal new physics.

Common Pitfalls

Confusing Feynman diagrams with physical trajectories. Feynman diagrams are calculational tools, not pictures of particle paths. Fix: Each diagram represents a term in the perturbation series; internal lines are virtual particles.
Wrong conservation law application. In particle reactions, conserve energy, momentum, charge, lepton number, baryon number, and strangeness (for strong interactions). Fix: Weak interactions can change strangeness; strong and EM interactions conserve it.
Confusing redshift types. Cosmological redshift: due to expansion of space. Doppler redshift: due to relative motion. Fix: For distant galaxies, cosmological redshift dominates; $z \approx H_0 d/c$ for small $z$ .

Summary

Standard Model: quarks, leptons, gauge bosons, Higgs boson; four fundamental forces.
Conservation laws: energy, momentum, charge, baryon number, lepton number, strangeness (strong/EM only).
Hubble’s law: $v = H_0 d$ ; evidence for the expanding universe.
Big Bang: CMB radiation, nucleosynthesis, dark matter and dark energy.

Cross-References

Topic	Site	Link
[Particle Physics]	A-Level	View
[Particle Physics]	University	View