One-Loop Electron Self-Energy and Renormalization

Hello! Let's dive into our next lesson on Quantum Field Theory.

In our last session, we successfully tackled the one-loop correction to the photon propagator, a process called vacuum polarization. We saw how dimensional regularization allowed us to handle the ultraviolet divergence, and how the concept of renormalization absorbed this infinity into a redefinition of the electric charge. The beautiful result was the prediction of a "running" coupling constant, a cornerstone of QED.

Today, we will apply this powerful framework to the electron itself. The learning outcome for this lesson is to calculate the one-loop electron self-energy and renormalize the electron mass and field strength. Just as quantum effects modify the force carrier (the photon), they also modify the properties of the matter particle (the electron). We'll discover that the "bare" mass and field that appear in our initial Lagrangian are not the physical quantities we observe in experiments.

1. The Dressed Propagator and the Self-Energy

The object we are interested in is the full, or "dressed," electron propagator, $S_{F}^{'} (p)$ . It represents the propagation of an electron from one point to another, including all possible quantum fluctuations. The simplest fluctuation is an electron emitting and reabsorbing a virtual photon.

One-Loop Electron Self-Energy Feynman Diagram — A Feynman diagram representing the one-loop electron self-energy. An electron (solid line) with momentum $p$ propagates, emits a virtual photon (wavy line), and reabsorbs it. This loop correction modifies the electron's properties.

Just as with the photon propagator, the full propagator can be expressed as a geometric series of the bare propagator, $S_{F} (p)$ , and the sum of all one-particle-irreducible (1PI) loop corrections. A diagram is 1PI if it cannot be split into two separate pieces by cutting a single internal propagator line. The sum of all such 1PI diagrams for the electron is called the electron self-energy, denoted $- i Σ (p)$ .

This resummation gives us a compact and powerful expression for the dressed propagator:

S_{F}^{'} (p) = S_{F} (p) + S_{F} (p) (- i Σ (p)) S_{F} (p) + S_{F} (p) (- i Σ (p)) S_{F} (p) (- i Σ (p)) S_{F} (p) + \dots

Summing this geometric series gives:

S_{F}^{'} (p) = \frac{1}{( S _{F} ( p ) ) ^{- 1} + i Σ ( p )} = \frac{1}{\frac{i ( \neq p - m _{0} )}{i ^{2}} + i Σ ( p )} = \frac{i}{\neq p - m _{0} - Σ ( p )}

Here, $m_{0}$ is the "bare" mass parameter from our original QED Lagrangian. This equation is profound: it shows that the quantum corrections contained in $Σ (p)$ shift the pole of the propagator. The physical mass, which corresponds to the location of the pole, will no longer be $m_{0}$ .

The following video provides an excellent conceptual introduction to the self-energy and the dressed propagator using a scalar field, which illustrates the general principles.

Field and Mass Renormalization - QFT II, Part 5

To build a solid conceptual foundation, please watch the following video segment. It explains the idea of a dressed propagator, one-particle-irreducible diagrams, and the self-energy in the context of a general scalar field theory. The logic is directly applicable to the electron in QED.

Watch the video 'Field and Mass Renormalization' from 01:45 to 23:04. Focus on: The distinction between amputated diagrams and full propagators. The classification of loop diagrams into one-particle-irreducible (1PI) and reducible ones. The definition of the self-energy (called M²(p²) in the video) as the sum of all 1PI diagrams. The geometric series resummation that leads to the final expression for the full propagator: i / (p² - m₀² - Σ(p²)).

2. Calculation of the One-Loop Electron Self-Energy

Now, let's calculate $Σ (p)$ for the electron at one-loop order in QED. Applying the Feynman rules to the diagram above, we get the integral:

- i Σ (p) = \int \frac{d ^{4} k}{( 2 π ) ^{4}} (- i e γ^{μ}) \frac{i ( \neq p - \neq k + m _{0} )}{( p - k ) ^{2} - m _{0}^{2} + i ϵ} (- i e γ_{μ}) \frac{- i}{k ^{2} - λ ^{2} + i ϵ}

We are working in Feynman gauge for the photon propagator and have given the photon a small fictitious mass $λ$ to regulate potential infrared (IR) divergences, which arise from massless particles. By power-counting, this integral is logarithmically divergent in the ultraviolet (UV), just like the vacuum polarization loop.

The calculation proceeds through the standard steps of dimensional regularization.

One-Loop Fermion Self-energy

Let's work through the explicit calculation. The following resource provides a concise, step-by-step derivation of the one-loop electron self-energy integral using dimensional regularization.

Please read section 3.1 'Fermion Self-energy' up to, but not including, the discussion of the infrared divergence that starts with 'This is a new type of divergence...' (i.e., stop before that paragraph on page 24). Focus on these steps: Setting up the integral: See how the Feynman rules are applied in d dimensions (eq. 3.1). Feynman Parametrization and Momentum Shift: Follow the combination of denominators and the shift in loop momentum k (eq. 3.2 to 3.3). Wick Rotation and Integration: The resource performs the d-dimensional momentum integral, leading to eq. 3.4. Expansion in ε: Observe how the result is expanded in ε to isolate the 1/ε pole and finite terms (eq. 3.5). Taylor Expansion: The most crucial step is the expansion of Σ(p) in a power series of (γ·p - m) (eq. 3.7). This separates the parts that will renormalize the mass from those that will renormalize the field strength.

After a considerable amount of algebra, the self-energy $Σ (p)$ can be expanded near the physical mass shell (i.e., for $\neq p \approx m$ ). The result from the reading takes the form:

Σ (p) \approx A + B (\neq p - m) + finite terms vanishing as \neq p \to m

Here, both $A$ and $B$ contain a divergent $1/ ϵ$ piece. This structure is exactly what we need to renormalize both the mass and the field.

3. Mass and Wavefunction Renormalization

The divergences in $Σ (p)$ mean our theory, as it stands, gives infinite corrections to the electron's properties. The solution, as before, is renormalization. We assert that the parameters in our initial Lagrangian are unobservable "bare" quantities ( $ψ_{0}, m_{0}, e_{0}$ ). We then redefine them in terms of physical, "renormalized" quantities ( $ψ, m, e$ ) and divergent counterterms.

The QED Lagrangian is rewritten as:

L = ψ ˉ_{0} (i \neq \partial - m_{0}) ψ_{0} - e_{0} ψ ˉ_{0} \neq A_{0} ψ_{0} - \frac{1}{4} F_{0, μν} F_{0}^{μν}

We define the renormalized fields and parameters:

$ψ_{0} = Z_{2}^{1/2} ψ$ (Wavefunction renormalization)
$m_{0} = Z_{m} m$ (Mass renormalization)
$A_{0, μ} = Z_{3}^{1/2} A_{μ}$ (Field renormalization, as before)
$e_{0} = Z_{e} e$ (Charge renormalization)

This allows us to split the Lagrangian into a renormalized part and a counterterm part:

L = L_{ren} ψ ˉ (i \neq \partial - m) ψ - e ψ ˉ \neq A ψ - \frac{1}{4} F_{μν} F^{μν} + L_{counterterm} δ Z_{2} ψ ˉ i \neq \partial ψ - δ Z_{m} m ψ ˉ ψ - eδ Z_{1} ψ ˉ \neq A ψ - \frac{1}{4} δ Z_{3} F_{μν} F^{μν}

where $Z_{i} = 1 + δ Z_{i}$ . The counterterms $δ Z_{2}$ and $δ Z_{m}$ are new. Their role is to precisely cancel the divergences arising from the electron self-energy.

The On-Shell Renormalization Scheme

How do we fix the finite parts of these counterterms? A physically intuitive method is the on-shell scheme. We demand that the renormalized parameters correspond to the properties of a real, physical electron.

Mass Renormalization: The physical mass $m$ is the pole of the full propagator. We enforce this by choosing $δ Z_{m}$ such that the full propagator $S_{F}^{'} (p)$ has a pole at $\neq p = m$ .
Wavefunction Renormalization: For a free particle, the residue of the propagator at the pole is $i$ . We demand the same for our interacting theory. We choose $δ Z_{2}$ so that the residue of $S_{F}^{'} (p)$ at $\neq p = m$ is exactly $i$ .

These two conditions ensure that our renormalized field $ψ$ creates and annihilates physical electron states with mass $m$ .

The following resources explain how this scheme is implemented. The video continues its excellent conceptual explanation, and the text formalizes the specific definitions for QED.

Field and Mass Renormalization - QFT II, Part 5

This video segment explains how to implement these physical conditions by introducing counterterms into the Lagrangian. It shows that the counterterms are simply a consequence of rewriting the theory in terms of physical, measurable quantities.

Please watch from 23:04 to 50:17. This part covers: Redefining the field and mass, and rewriting the Lagrangian to isolate the counterterms ΔZ and Δm (23:04 - 34:31). Deriving the explicit on-shell conditions that fix these counterterms: the self-energy and its derivative must vanish at the pole (34:31 - 50:17).

On-shell renormalization of electron mass and field

The on-shell scheme provides a physically intuitive way to define our renormalized parameters. This resource formalizes the procedure we've discussed specifically for the electron propagator in QED.

Please read section 6.3, 'On-shell renormalization of electron mass and field' (pages 69-70). Pay close attention to: How the Lagrangian is rewritten with a mass counterterm δm (eq. 6.13). The definition of the physical mass as the pole of the full propagator (eq. 6.18-6.19). The definition of the on-shell field ψ_os such that the propagator has residue i at the pole (eq. 6.22-6.24). The final result for the one-loop counterterms Z_m^os and Z_ψ^os (eq. 6.29), which are derived from the calculation we saw earlier.

In essence, the divergent term $A$ from our calculation of $Σ (p)$ is cancelled by the mass counterterm $δ Z_{m} m$ , and the divergent term $B$ is cancelled by the wavefunction counterterm $δ Z_{2}$ . All UV infinities are once again absorbed into unobservable bare quantities, leaving a finite, predictive theory.

Physical Interpretation

Mass Renormalization: The physical electron is constantly interacting with the quantum vacuum, emitting and reabsorbing virtual photons. The energy of this interaction cloud contributes to the electron's total energy, and therefore its inertia, which is what we measure as mass. The bare mass $m_{0}$ is a fictional quantity; the physical mass $m$ is the bare mass "dressed" by its self-interactions.
Wavefunction Renormalization: The constant $Z_{2}$ is the probability amplitude of finding a "bare" electron state within the "dressed," physical electron state. Since the physical electron is a superposition of the bare electron plus states with virtual photons, this probability is less than one ( $Z_{2} < 1$ ).

Test your understanding!

In the on-shell scheme, we defined our counterterms $δ m$ and $δ Z_{2}$ to ensure the renormalized propagator has a pole at the physical mass $m$ with residue $i$ .

Suppose you were working in the MS-bar ( $MS$ ) scheme instead. Based on what you learned about the $MS$ scheme in the last lesson, how would you define the counterterms $δ m$ and $δ Z_{2}$ ? Would the resulting renormalized self-energy $Σ_{R} (p)$ be zero when $\neq p = m$ ?

Show answer

In the $MS$ scheme, the counterterms are chosen to cancel only the $1/ ϵ$ pole and the associated universal constants $ln (4 π) - γ_{E}$ , regardless of the on-shell behavior.

Looking at the result from Resource LINK (eq. 3.7), the calculation separates the divergent pieces that are constant and linear in $(\neq p - m)$ . In the $MS$ scheme:

$δ m_{M S}$ would be chosen to cancel the divergent part of the constant term (the part proportional to $3 m \frac{α}{4 π} \frac{1}{ϵ}$ ).
$(Z_{2} - 1)_{M S}$ would be chosen to cancel the divergent part of the coefficient of $(\neq p - m)$ (the part proportional to $- \frac{α}{4 π} \frac{1}{ϵ}$ ).

Because these counterterms only subtract the pole part, the remaining renormalized self-energy, $Σ_{R} (p)$ , will not be zero at $\neq p = m$ . The $MS$ scheme is computationally simpler but breaks the direct connection between the parameter $m$ in the Lagrangian and the physical pole mass. The two are related by a finite, calculable correction.

Conclusion

Today, we have extended our understanding of renormalization to matter fields. We saw how the electron's self-interaction with virtual photons leads to divergences in its mass and propagator residue.

Key Takeaways:

The full electron propagator can be expressed in terms of the electron self-energy $Σ (p)$ , which is the sum of all 1PI loop corrections.
The one-loop self-energy integral is UV divergent and requires regularization.
Mass renormalization accounts for the fact that the physical mass of the electron includes its self-interaction energy. We introduce a mass counterterm $δ Z_{m} m$ to absorb the corresponding divergence.
Wavefunction renormalization ensures that the physical electron state is correctly normalized. We introduce a kinetic counterterm $δ Z_{2}$ to absorb the divergence in the propagator residue.
In the on-shell scheme, counterterms are fixed by requiring the renormalized propagator to have a pole at the physical mass $m$ with residue $i$ , making a direct link to observable quantities.

Preview of the Next Lesson:
We have now dealt with two of the three "primitive" divergences of QED: vacuum polarization and electron self-energy. There is one more: the vertex correction. In our next lesson, we will calculate the one-loop correction to the QED interaction vertex itself. This will complete our one-loop picture of QED and lead to one of its most celebrated predictions: the anomalous magnetic moment of the electron.

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