Quantum Field Theory Course

Total: 440+ pages covering classical field theory through the Standard Model

Part IV, Chapter 5 | Page 4 of 4

Physical Applications and Precision Tests

R-ratio, anomalous magnetic moment, and the running of the fine-structure constant

5.13 The R-Ratio: Probing QCD with QED

One of the most important applications of $e^+e^-$ annihilation physics is the R-ratio, defined as the ratio of the hadronic cross section to the muon pair production cross section:

$$R(s) \equiv \frac{\sigma(e^+e^- \to \text{hadrons})}{\sigma(e^+e^- \to \mu^+\mu^-)}$$

The denominator is the pure QED cross section we computed:

$$\sigma(e^+e^- \to \mu^+\mu^-) = \frac{4\pi\alpha^2}{3s}$$

At energies well above quark thresholds but below $Z$-pole effects, the numerator is computed by summing over all kinematically accessible quark flavors. Each quark flavor$q$ with electric charge $Q_q$ contributes like a lepton pair but with color multiplicity $N_c = 3$:

$$\boxed{R = N_c \sum_q Q_q^2 = 3 \sum_q Q_q^2}$$

Evaluating for different energy ranges:

  • Below charm threshold ($\sqrt{s} < 2m_c$): $R = 3\left[\left(\frac{2}{3}\right)^2 + \left(\frac{1}{3}\right)^2 + \left(\frac{1}{3}\right)^2\right] = 3 \cdot \frac{6}{9} = 2$
  • Above charm, below bottom ($2m_c < \sqrt{s} < 2m_b$): $R = 3\left[\frac{6}{9} + \frac{4}{9}\right] = \frac{10}{3} \approx 3.33$
  • Above bottom ($\sqrt{s} > 2m_b$): $R = 3\left[\frac{6}{9} + \frac{4}{9} + \frac{1}{9}\right] = \frac{11}{3} \approx 3.67$

QCD corrections modify the leading-order result:

$$R = N_c \sum_q Q_q^2 \left(1 + \frac{\alpha_s(s)}{\pi} + \mathcal{O}(\alpha_s^2)\right)$$

💡R-Ratio as a Color Counter

The factor of $N_c = 3$ in the R-ratio is direct experimental evidence that quarks come in three colors. Without color, the prediction $R = 2/3$ below charm threshold would be three times too small compared to the measured value of$R \approx 2$. The step-like structure of R as new quark thresholds are crossed provides a beautiful counting of quark degrees of freedom.

5.14 The Anomalous Magnetic Moment

The most precise test of QED — and arguably of any quantum field theory — is the anomalous magnetic moment of the electron. The Dirac equation predicts$g = 2$ for a spin-1/2 particle. Quantum loop corrections shift this to:

$$g = 2(1 + a), \qquad a \equiv \frac{g-2}{2}$$

Schwinger's famous one-loop result (1948) gives the leading correction from the vertex diagram where a virtual photon connects the two fermion legs:

$$\boxed{a_e^{(1)} = \frac{\alpha}{2\pi} \approx 0.00116}$$

Higher-order QED contributions have been computed to extraordinary precision. The current theoretical prediction includes terms through fifth order in $\alpha$:

$$a_e^{\text{QED}} = \frac{\alpha}{2\pi} - 0.328\,478\,965\ldots\left(\frac{\alpha}{\pi}\right)^2 + 1.181\,241\,456\ldots\left(\frac{\alpha}{\pi}\right)^3 - \cdots$$

The comparison between theory and experiment for the electron:

$$a_e^{\text{exp}} = 0.001\,159\,652\,180\,73(28)$$
$$a_e^{\text{theory}} = 0.001\,159\,652\,182\,032(720)$$

Agreement to better than one part in $10^{10}$ — the most precisely tested prediction in all of physics.

The muon anomalous magnetic moment

The muon $g-2$ is more sensitive to new physics because heavy virtual particles contribute with amplitudes scaling as $(m_\mu/m_e)^2 \approx 43{,}000$. The current experimental status shows a tantalizing discrepancy:

$$\Delta a_\mu = a_\mu^{\text{exp}} - a_\mu^{\text{SM}} \approx (249 \pm 48) \times 10^{-11}$$

representing a $\sim 5\sigma$ tension (depending on the hadronic vacuum polarization calculation used), which may indicate physics beyond the Standard Model or could be resolved by improved lattice QCD calculations of hadronic contributions.

💡Why g-2 Is So Powerful

The anomalous magnetic moment probes every sector of physics that couples to the electron or muon. The QED contribution dominates, but hadronic (strong-force) and electroweak corrections also contribute measurably. Any new particle with mass M coupling to leptons would contribute $\delta a \sim (m_\ell/M)^2$, making g-2 sensitive to virtual effects from particles too heavy to produce directly.

5.15 The Running of $\alpha$

The fine-structure constant is not actually constant — it runs with the energy scale due to vacuum polarization. Virtual $e^+e^-$ pairs screen the bare electric charge, reducing its effective value at large distances (low energies).

At one loop, the photon propagator receives a correction from the vacuum polarization diagram (fermion loop insertion). The renormalized coupling at momentum transfer$q^2$ is:

$$\alpha(q^2) = \frac{\alpha(0)}{1 - \Delta\alpha(q^2)}$$

where the one-loop vacuum polarization contribution from a fermion of mass $m_f$and charge $Q_f$ is:

$$\Delta\alpha_f(q^2) = -\frac{\alpha Q_f^2}{3\pi}\left[\ln\frac{|q^2|}{m_f^2} - \frac{5}{3} + \mathcal{O}\!\left(\frac{m_f^2}{q^2}\right)\right] \qquad (|q^2| \gg m_f^2)$$

The running can be equivalently expressed via the QED beta function:

$$\mu\frac{d\alpha}{d\mu} = \beta(\alpha) = \frac{2\alpha^2}{3\pi}\sum_f N_c^f Q_f^2 + \mathcal{O}(\alpha^3)$$

Note the positive beta function: $\alpha$ increases at higher energies. Summing over all Standard Model fermions, the value at the Z-pole is:

$$\alpha(0) = \frac{1}{137.036}, \qquad \alpha(m_Z^2) = \frac{1}{128.9}$$

This 6% increase from $q^2 = 0$ to $q^2 = m_Z^2$ is experimentally confirmed and essential for precision electroweak physics.

💡Screening vs. Anti-screening

In QED, the vacuum polarization cloud screens the bare charge: at large distances you see a reduced effective charge. At short distances (high $q^2$), you penetrate the screening cloud and see more of the bare charge, so $\alpha$increases. This is opposite to QCD, where gluon self-interactions cause anti-screening, making $\alpha_s$ decrease at high energies (asymptotic freedom). This fundamental difference explains why QCD confines at low energies while QED does not.

5.16 Chapter Summary

In this chapter we have explored the rich landscape of tree-level and one-loop QED processes. The key themes that emerged are:

  • Identical particles: Moller scattering demonstrates that identical fermions require antisymmetrized amplitudes, with the relative minus sign encoding Fermi statistics
  • Crossing symmetry: A single analytic amplitude function describes multiple physical processes, related by continuation in Mandelstam variables
  • Quantum vacuum effects: Photon-photon scattering and vacuum birefringence arise from virtual fermion loops, revealing that the vacuum is a dynamical medium
  • Precision tests: QED predictions for g-2 agree with experiment to parts per billion, validating the framework of perturbative quantum field theory
  • Running couplings: The fine-structure constant is not constant but evolves logarithmically with energy scale, a consequence of vacuum polarization

These results set the stage for systematic loop calculations and renormalization, which we develop in the following chapters.

Key Concepts (Page 4)

  • • R-ratio: $R = N_c \sum_q Q_q^2$ directly counts quark colors and flavors
  • • Schwinger result: $a_e = \alpha/(2\pi)$ — first quantum correction to g = 2
  • • Electron g-2 verified to $10^{-10}$ precision — most precise test in physics
  • • Muon g-2 shows $\sim 5\sigma$ tension with Standard Model (depending on hadronic input)
  • • $\alpha$ runs from 1/137 at $q^2 = 0$ to 1/129 at $q^2 = m_Z^2$ due to vacuum polarization
  • • QED has a positive beta function (charge screening); opposite to QCD (asymptotic freedom)
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