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