Quantum Field Theory Course

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Part IV, Chapter 6 | Page 3 of 4

Electron Self-Energy

Mass renormalization, wavefunction renormalization, and the physical electron

6.12 The One-Loop Electron Self-Energy

The electron self-energy arises from the emission and reabsorption of a virtual photon. This one-loop correction modifies the electron propagator. The self-energy function$-i\Sigma(\not\!p)$ is:

$$-i\Sigma(\not\!p) = (-ie)^2 \int \frac{d^dk}{(2\pi)^d} \frac{\gamma^\mu(\not\!p - \not\!k + m)\gamma_\mu}{[(p-k)^2 - m^2 + i\epsilon][k^2 - \mu_\gamma^2 + i\epsilon]}$$

where $\mu_\gamma$ is a fictitious photon mass regulating infrared divergences. By Lorentz covariance, $\Sigma$ can only depend on $\not\!p$ and $m$:

$$\Sigma(\not\!p) = A(p^2)\not\!p + B(p^2) m$$

or equivalently, decomposing near the mass shell $\not\!p = m$:

$$\Sigma(\not\!p) = \Sigma(m) + (\not\!p - m)\Sigma'(m) + \hat\Sigma(\not\!p)$$

where $\Sigma(m) \equiv \Sigma(\not\!p)\big|_{\not\!p=m}$,$\Sigma'(m) \equiv \frac{\partial\Sigma}{\partial\not\!p}\big|_{\not\!p=m}$, and $\hat\Sigma$ vanishes at $\not\!p = m$ along with its first derivative.

💡The Electron's Photon Cloud

A physical electron is never "bare" — it is always surrounded by a cloud of virtual photons. The self-energy $\Sigma(\not\!p)$ encodes how this cloud modifies the electron's propagation. The cloud changes the effective mass ($\Sigma(m)$) and the normalization of the electron field ($\Sigma'(m)$). Only the finite remainder $\hat\Sigma$ produces new momentum-dependent physics.

6.13 Evaluating $\Sigma(\not\!p)$

Step 1: Dirac algebra. Contract the gamma matrices in $d$ dimensions using $\gamma^\mu \gamma^\nu \gamma_\mu = -(d-2)\gamma^\nu$:

$$\gamma^\mu(\not\!p - \not\!k + m)\gamma_\mu = -(d-2)(\not\!p - \not\!k) + dm$$

In $d = 4 - \varepsilon$ dimensions, this becomes$-(2-\varepsilon)(\not\!p - \not\!k) + (4-\varepsilon)m$.

Step 2: Feynman parametrization. Combine the denominators:

$$\frac{1}{[(p-k)^2 - m^2][k^2 - \mu_\gamma^2]} = \int_0^1 dx\;\frac{1}{[\ell^2 - \Delta]^2}$$

with the shift $k \to \ell = k - xp$ and:

$$\Delta = -x(1-x)p^2 + xm^2 + (1-x)\mu_\gamma^2$$

Step 3: Momentum integration. The numerator after shifting contains a term linear in $\ell$ (which vanishes under symmetric integration) and a constant piece. Using the standard dimensional regularization integral:

$$\int \frac{d^d\ell}{(2\pi)^d} \frac{1}{(\ell^2 - \Delta)^2} = \frac{i}{(4\pi)^{d/2}}\frac{\Gamma(2 - d/2)}{\Delta^{2-d/2}}$$

The full result is:

$$\Sigma(\not\!p) = \frac{\alpha}{4\pi}\int_0^1 dx\;\left[-(2-\varepsilon)(1-x)\not\!p + (4-\varepsilon)m\right]\left[\frac{2}{\varepsilon} - \gamma_E + \ln(4\pi) - \ln\frac{\Delta}{\mu^2}\right]$$

Separating into divergent and finite parts:

$$\boxed{\Sigma(\not\!p) = \frac{\alpha}{4\pi}\left(\frac{2}{\varepsilon}\right)\left[-\not\!p + 4m\right] + \text{finite}(p^2)}$$

The self-energy contains both a mass-type divergence (proportional to $m$) and a wavefunction-type divergence (proportional to $\not\!p$).

6.14 Mass Renormalization

The full electron propagator, including the self-energy, is obtained by summing the geometric series of 1PI insertions:

$$S_F(p) = \frac{i}{\not\!p - m_0 - \Sigma(\not\!p) + i\epsilon}$$

The physical (pole) mass $m$ is defined as the location of the propagator pole:

$$\not\!p - m_0 - \Sigma(\not\!p)\big|_{\not\!p = m} = 0 \quad \Longrightarrow \quad m = m_0 + \Sigma(m)$$

Defining the mass counterterm $\delta m = \Sigma(m)$:

$$\boxed{\delta m = \Sigma(m) = \frac{3\alpha}{4\pi}m\left[\frac{2}{\varepsilon} - \gamma_E + \ln\frac{4\pi\mu^2}{m^2} + \frac{4}{3} + O\!\left(\frac{\mu_\gamma^2}{m^2}\right)\right]}$$

The mass correction is proportional to $m$ itself: a massless electron receives no mass correction. This is a consequence of chiral symmetry — if $m = 0$, the QED Lagrangian has a chiral symmetry that forbids mass generation at any order in perturbation theory.

💡Bare Mass vs. Physical Mass

The bare mass $m_0$ in the Lagrangian is not the mass you measure. The measured mass $m$ includes the energy stored in the photon cloud surrounding the electron. Formally, $m_0$ is divergent (it must be, to cancel the divergence in $\delta m$), but $m = m_0 + \delta m$ is the finite, physical quantity. We never need to know$m_0$ separately — only $m$ appears in predictions.

6.15 Wavefunction Renormalization $Z_2$

Near the mass shell, the dressed propagator behaves as:

$$S_F(p) = \frac{i}{\not\!p - m - \Sigma(\not\!p) + \Sigma(m)} \approx \frac{iZ_2}{\not\!p - m + i\epsilon} \quad \text{as } \not\!p \to m$$

where the wavefunction renormalization constant $Z_2$ is the residue of the pole:

$$Z_2^{-1} = 1 - \Sigma'(m) = 1 - \frac{\partial\Sigma}{\partial\not\!p}\bigg|_{\not\!p = m}$$

Evaluating the derivative of $\Sigma$ at the mass shell:

$$\boxed{Z_2 = 1 + \frac{\alpha}{4\pi}\left[-\frac{2}{\varepsilon} + \gamma_E - \ln\frac{4\pi\mu^2}{m^2} - 4 + 3\ln\frac{m^2}{\mu_\gamma^2}\right] + O(\alpha^2)}$$

Note that $Z_2$ contains both a UV divergence ($1/\varepsilon$) and an IR divergence ($\ln\mu_\gamma$). The UV divergence is canceled by the counterterm$\delta Z_2 = Z_2 - 1$. The IR divergence will cancel in physical cross sections when combined with soft-photon bremsstrahlung (as we discuss on Page 4).

💡What Does Z_2 Mean Physically?

$Z_2$ relates the bare and renormalized electron fields:$\psi_0 = \sqrt{Z_2}\,\psi_R$. Physically, it accounts for the probability that the physical electron is found in the "bare" one-particle state. Since the physical electron has a nonzero probability of being accompanied by virtual photons,$Z_2 < 1$ — the bare-electron component is reduced. The factor $\sqrt{Z_2}$corrects the normalization of external fermion lines in S-matrix elements.

6.16 On-Shell Renormalization Conditions

The on-shell renormalization scheme defines physical parameters by requiring that the propagator has the standard form near the mass shell. For the electron:

On-Shell Conditions for the Electron Propagator

Condition 1 (Mass): The pole of $S_F(p)$ is at $p^2 = m^2$, where $m$ is the physical mass:
$$\Sigma(m) = \delta m \quad \Longleftrightarrow \quad \hat\Sigma(m) = 0$$
Condition 2 (Residue): The residue of the pole equals $i$:
$$\frac{\partial\hat\Sigma}{\partial\not\!p}\bigg|_{\not\!p = m} = 0 \quad \Longleftrightarrow \quad Z_2 = 1 + \delta Z_2$$

With these conditions, the renormalized propagator near the mass shell takes the standard form:

$$S_F^{\text{ren}}(p) = \frac{i}{\not\!p - m - \hat\Sigma(\not\!p)} \approx \frac{i}{\not\!p - m} + O\!\left((\not\!p - m)^0\right) \quad\text{as } \not\!p \to m$$

The complete set of on-shell conditions for QED renormalization at one loop is:

\begin{align*} \text{Electron mass:}&\quad \hat\Sigma(m) = 0 \\ \text{Electron residue:}&\quad \hat\Sigma'(m) = 0 \\ \text{Photon residue:}&\quad \hat\Pi(0) = 0 \\ \text{Vertex:}&\quad \hat\Gamma^\mu(p',p)\big|_{p'=p,\,\not\!p=m} = \gamma^\mu \end{align*}

These four conditions fix the four counterterms $\delta m$, $\delta Z_2$,$\delta Z_3$, and $\delta Z_1$ (or equivalently $\delta e$). After imposing them, all remaining predictions of QED at one loop (and beyond) are finite and unambiguous.

6.17 The Renormalized Self-Energy

After subtracting the counterterms, the renormalized (finite) self-energy is:

$$\hat\Sigma(\not\!p) = \Sigma(\not\!p) - \delta m - (\not\!p - m)\delta Z_2$$

Explicitly, the finite remainder is:

$$\hat\Sigma(\not\!p) = \frac{\alpha}{4\pi}\int_0^1 dx\;\left[-(2-\varepsilon)(1-x)\not\!p + (4-\varepsilon)m\right]\ln\!\left(\frac{\Delta(p^2)}{\Delta(m^2)}\right) + O((\not\!p - m)^2)$$

This vanishes at $\not\!p = m$ and its derivative vanishes there too, as required by the renormalization conditions. The physical content is the momentum dependence away from the mass shell, which contributes to off-shell processes and higher-order diagrams.

In practice, the electron self-energy is most important through its effect on $Z_2$in the LSZ reduction formula: external fermion lines in S-matrix elements carry a factor of $\sqrt{Z_2}$, which modifies cross sections at order $\alpha$ relative to tree level.

Key Concepts (Page 3)

  • • The electron self-energy $\Sigma(\not\!p)$ modifies both the mass and field normalization
  • • Feynman parametrization + dim. reg. yields UV-divergent pieces $\propto \not\!p/\varepsilon$ and $\propto m/\varepsilon$
  • • Mass counterterm: $\delta m = \Sigma(m) \propto \alpha m/\varepsilon$ (protected by chiral symmetry if $m = 0$)
  • • Wavefunction renormalization: $Z_2 = 1/(1 - \Sigma'(m))$ contains both UV and IR divergences
  • • On-shell conditions fix $\delta m$ and $\delta Z_2$ uniquely
  • • The physical propagator $\to i/(\not\!p - m)$ near the mass shell with unit residue
  • • External leg corrections enter physical amplitudes via the LSZ formula through $\sqrt{Z_2}$
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