Quantum Field Theory Course

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

Part IV, Chapter 6 | Page 4 of 4

Renormalizability, Ward Identity & g-2

UV and IR divergences, the structure of QED renormalization, and precision tests

6.18 UV and IR Divergences: A Complete Picture

QED loop diagrams contain two distinct types of divergence, which must be handled by different mechanisms:

Ultraviolet (UV) Divergences

Arise from loop momenta $k \to \infty$. Present in the three primitive diagrams:

  • Vertex correction: logarithmic ($D = 0$)
  • Electron self-energy: linear by power counting ($D = 1$), but reduced to logarithmic by Dirac structure
  • Vacuum polarization: quadratic by power counting ($D = 2$), but reduced to logarithmic by gauge invariance

Resolution: Absorbed into counterterms $\delta m$, $\delta Z_2$,$\delta Z_3$, $\delta Z_1$ via renormalization.

Infrared (IR) Divergences

Arise from loop momenta $k \to 0$ (soft photons) or from photons collinear with massless fermions. Present in:

  • Vertex correction: $F_1(q^2)$ contains $\ln(\mu_\gamma/m)$
  • Electron self-energy: $Z_2$ contains $\ln(m/\mu_\gamma)$

Resolution: The Bloch-Nordsieck theorem guarantees that IR divergences cancel in inclusive cross sections when real soft-photon emission (bremsstrahlung) is included.

The cancellation of IR divergences is a deep physical requirement: a detector with finite energy resolution cannot distinguish an electron from an electron accompanied by an arbitrarily soft photon. The inclusive rate (virtual correction + real emission) must therefore be finite:

$$\sigma_{\text{phys}} = \sigma_{\text{virtual}}(\mu_\gamma) + \sigma_{\text{brem}}(\mu_\gamma, E_{\min}) = \text{finite}$$

where $E_{\min}$ is the detector's minimum photon energy resolution. The$\mu_\gamma$ dependence cancels between the two terms.

💡Why IR Divergences Must Cancel

Imagine measuring $e^-\mu^- \to e^-\mu^-$ scattering. Your detector cannot see photons below some energy threshold $E_{\min}$. The "elastic" rate you measure actually includes events $e^-\mu^- \to e^-\mu^-\gamma$ where the photon is too soft to detect. The virtual loop correction (with its IR divergence) plus the undetected real emission (also IR divergent) must give a finite, physical answer that depends on $E_{\min}$ but not on the fictitious regulator $\mu_\gamma$.

6.19 Renormalizability of QED

A theory is renormalizable if all UV divergences at every loop order can be absorbed into a finite number of counterterms corresponding to parameters already present in the Lagrangian. For QED, this means four counterterms suffice to all orders:

$$\delta m, \quad \delta Z_1, \quad \delta Z_2, \quad \delta Z_3$$

The proof relies on the superficial degree of divergence. For a QED diagram with $E_f$ external fermion lines and $E_\gamma$ external photon lines:

$$D = 4 - \frac{3}{2}E_f - E_\gamma$$

Crucially, $D$ depends only on the external structure, not on the number of loops. This means that at any loop order, the same set of external-leg structures can produce divergences. Since there are finitely many structures with $D \geq 0$, only finitely many counterterms are needed.

The structures with $D \geq 0$ are:

\begin{align*} (E_f, E_\gamma) &= (0, 2): &\quad D &= 2 & &\text{(photon self-energy)} \\ (E_f, E_\gamma) &= (0, 4): &\quad D &= 0 & &\text{(light-by-light, vanishes by Furry)} \\ (E_f, E_\gamma) &= (2, 0): &\quad D &= 1 & &\text{(electron self-energy)} \\ (E_f, E_\gamma) &= (2, 1): &\quad D &= 0 & &\text{(vertex correction)} \\ (E_f, E_\gamma) &= (0, 1): &\quad D &= 3 & &\text{(photon tadpole, vanishes by Furry)} \\ (E_f, E_\gamma) &= (0, 3): &\quad D &= 1 & &\text{(3-photon vertex, vanishes by Furry)} \end{align*}

Furry's theorem eliminates diagrams with an odd number of photon vertices (they vanish by charge conjugation symmetry). The four-photon vertex ($D = 0$) is convergent once gauge invariance is imposed (it actually has $D_{\text{eff}} = -4$). So only three structures genuinely diverge, requiring the four counterterms listed above.

6.20 The Ward Identity and Its Consequences

The Ward identity is the most important structural relation in QED. It connects the vertex function to the electron self-energy:

$$\boxed{q_\mu \Gamma^\mu(p', p) = S_F^{-1}(p') - S_F^{-1}(p)}$$

where $q = p' - p$. This is the quantum-mechanical generalization of classical current conservation $\partial_\mu J^\mu = 0$. At tree level it reduces to$q_\mu \gamma^\mu = (\not\!p' - m) - (\not\!p - m)$, which is trivially satisfied.

The Ward identity has profound consequences for renormalization:

$$\boxed{Z_1 = Z_2}$$

This means the vertex renormalization constant equals the electron wavefunction renormalization constant, to all orders in perturbation theory. The physical consequences are:

  • Charge universality: The physical charge $e$ is the same for all charged particles. Since $e = e_0 Z_1/(Z_2 \sqrt{Z_3}) = e_0/\sqrt{Z_3}$, the charge renormalization depends only on $Z_3$ (vacuum polarization), which is independent of the fermion species.
  • Charge conservation: The Ward identity ensures that radiative corrections do not alter the coupling strength at zero momentum transfer: $F_1(0) = 1$ exactly.
  • Reduced counterterms: Instead of four independent counterterms, we effectively have three: $\delta m$, $\delta Z_2$ (= $\delta Z_1$), and $\delta Z_3$.

💡Ward Identity = Gauge Invariance

The Ward identity is the perturbative manifestation of gauge invariance. It guarantees that the longitudinal (unphysical) polarization of the photon decouples from all physical processes. Without it, QED would predict different charges for different particles, violating the universality of electromagnetism. The identity holds order-by-order in perturbation theory and can be verified by explicit calculation at each loop order.

6.21 The Ward-Takahashi Identity

The full Ward-Takahashi identity applies to the off-shell vertex function and is the fundamental form from which all Ward identities derive. For the complete (1PI) vertex function:

$$q_\mu \Lambda^\mu(p', p) = \Sigma(p') - \Sigma(p)$$

where $\Lambda^\mu = \Gamma^\mu - \gamma^\mu$ is the loop correction to the vertex. Taking the limit $q \to 0$ (with $p' \to p$):

$$\Lambda^\mu(p, p) = \frac{\partial\Sigma(p)}{\partial p_\mu}$$

This directly implies $Z_1 = Z_2$: the UV-divergent parts of the vertex correction and the electron self-energy derivative are identical. This is verifiable by explicit calculation at one loop and is guaranteed to all orders by gauge symmetry.

6.22 The Anomalous Magnetic Moment: Precision QED

The anomalous magnetic moment $a_e = (g-2)/2$ is the most precisely tested prediction in all of physics. The vertex correction gives:

$$\Gamma^\mu(p', p) = F_1(q^2)\gamma^\mu + F_2(q^2)\frac{i\sigma^{\mu\nu}q_\nu}{2m}$$

The magnetic moment is related to the static limit of the Pauli form factor:

$$a_e = F_2(0)$$

The perturbative expansion in powers of $\alpha/\pi$ is:

$$a_e = \underbrace{\frac{1}{2}\frac{\alpha}{\pi}}_{\text{Schwinger}} - 0.32848\left(\frac{\alpha}{\pi}\right)^2 + 1.18124\left(\frac{\alpha}{\pi}\right)^3 - 1.9113\left(\frac{\alpha}{\pi}\right)^4 + 9.16\left(\frac{\alpha}{\pi}\right)^5 + \cdots$$

The coefficients at each order come from increasingly complex Feynman diagrams:

  • 1 loop (Schwinger 1948): 1 diagram, $C_1 = 0.5$
  • 2 loops (Petermann, Sommerfield 1957): 7 diagrams
  • 3 loops (Laporta, Remiddi 1996): 72 diagrams
  • 4 loops (Kinoshita et al. 2012): 891 diagrams, numerical evaluation
  • 5 loops (Aoyama et al. 2019): 12,672 diagrams, numerical evaluation

The current theoretical and experimental values:

\begin{align*} a_e^{\text{theory}} &= 0.001\,159\,652\,181\,78\,(77) \\ a_e^{\text{expt}} &= 0.001\,159\,652\,180\,73\,(28) \end{align*}

Agreement to better than one part in $10^{10}$. This is the most precise test of any physical theory in the history of science.

💡Why g-2 Is So Powerful

The anomalous magnetic moment is special because $F_2(0)$ is UV-finite at every order — it receives no divergent contributions and requires no renormalization. It is a direct, unambiguous prediction of QED (and the Standard Model). Any discrepancy between theory and experiment would signal new physics. The muon$g-2$, which is more sensitive to heavy new particles by a factor of$(m_\mu/m_e)^2 \approx 40000$, is currently the subject of intense experimental investigation at Fermilab.

6.23 The Muon g-2 and Hints of New Physics

For the muon, the anomalous magnetic moment receives contributions not only from QED but also from hadronic and electroweak loops:

$$a_\mu = a_\mu^{\text{QED}} + a_\mu^{\text{hadronic}} + a_\mu^{\text{EW}} + a_\mu^{\text{new physics?}}$$

The sensitivity to new heavy particles of mass $M$ scales as:

$$\delta a_\ell \sim \frac{\alpha}{\pi}\frac{m_\ell^2}{M^2}$$

This is why the muon is $(m_\mu/m_e)^2 \approx 43000$ times more sensitive to new physics than the electron. The Fermilab Muon g-2 experiment has measured:

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

representing a $\sim 5\sigma$ deviation from the Standard Model prediction (using data-driven hadronic vacuum polarization). However, the situation remains complex, as lattice QCD calculations of the hadronic contribution are still being refined.

6.24 Summary: The Complete One-Loop Structure of QED

Collecting all results from this chapter, the one-loop renormalization of QED is characterized by:

\begin{align*} \text{Mass:} \quad &\delta m = \frac{3\alpha}{4\pi}\,m\left(\frac{2}{\varepsilon} + \text{finite}\right) \\[8pt] \text{Electron field:} \quad &Z_2 = 1 - \frac{\alpha}{4\pi}\left(\frac{2}{\varepsilon} + \text{finite}\right) \\[8pt] \text{Photon field:} \quad &Z_3 = 1 - \frac{\alpha}{3\pi}\left(\frac{2}{\varepsilon} + \text{finite}\right) \\[8pt] \text{Vertex:} \quad &Z_1 = Z_2 \quad\text{(Ward identity)} \\[8pt] \text{Charge:} \quad &e_{\text{phys}} = \frac{e_0}{\sqrt{Z_3}} = e_0\left(1 + \frac{\alpha}{6\pi\varepsilon} + \cdots\right) \end{align*}

The fact that only $Z_3$ enters the charge renormalization (thanks to $Z_1 = Z_2$) means that the running of $\alpha$ is entirely determined by vacuum polarization — the screening of charge by virtual pairs. This is a beautiful and deep result of gauge invariance.

Key Concepts (Page 4)

  • • UV divergences are absorbed by renormalization; IR divergences cancel with bremsstrahlung in inclusive rates
  • • QED is renormalizable: the superficial degree of divergence $D = 4 - \frac{3}{2}E_f - E_\gamma$ limits divergent structures
  • • The Ward identity $Z_1 = Z_2$ follows from gauge invariance and ensures charge universality
  • • Charge renormalization depends only on $Z_3$: $e = e_0/\sqrt{Z_3}$
  • • $a_e = (g-2)/2 = \alpha/(2\pi) + \cdots$ is UV-finite and agrees with experiment to $10^{-10}$
  • • The muon $g-2$ is $(m_\mu/m_e)^2$ times more sensitive to new physics
  • • QED's precision predictions represent the greatest triumph of quantum field theory

Chapter 6 Complete

You have completed the Radiative Corrections & Renormalization chapter. You can now:

  • • Compute the one-loop vertex correction and extract the anomalous magnetic moment
  • • Evaluate the vacuum polarization and derive the running coupling
  • • Calculate the electron self-energy and perform mass and wavefunction renormalization
  • • Explain why QED is renormalizable and the role of the Ward identity
  • • Understand the interplay of UV and IR divergences in physical predictions

Continue to: Part V (Non-Abelian Gauge Theories) or Part VI (Advanced Renormalization) for the extension of these ideas to the full Standard Model.

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