Identifying UV Divergences in One-Loop QED Diagrams

1. From Trees to Loops: The Origin of Infinities

In all our calculations so far, the momenta of all internal particles were uniquely determined by the momenta of the external particles and the conservation of momentum at each vertex.

When we consider diagrams with closed loops, this is no longer the case. A particle in a loop has a momentum that is not constrained by the external kinematics. According to the principles of quantum mechanics, we must sum over all possibilities for this internal state. In a field theory, this "sum" becomes an integral over all possible values of the unconstrained loop momentum, from $0$ to $\infty$.

It is this integral over an infinite range of momenta that leads to trouble. To get a clear conceptual picture of this, let's start with a short video.

As the video explains, ultraviolet (UV) divergences come from the high-momentum (short-wavelength) region of the integration. Intuitively, these are probes of the theory's structure at infinitesimally small distances, where our current formulation breaks down.

2. Superficial Degree of Divergence

To make this more concrete, we can analyze the structure of these loop integrals. A typical one-loop integral in four spacetime dimensions has the form:

Here, $k$ is the unconstrained loop momentum. The "stuff" in the integrand consists of propagators (which go like $1/k^2$ or $1//k$ for large $k$) and vertex factors.

We can estimate whether an integral will diverge by simple "power counting" in the large-$k$ limit. This leads to the superficial degree of divergence, often denoted by $D$. For an integral over a loop momentum $k$, it's defined as:

The +4 comes from the integration measure $d^{4}k$. If $D\ge 0$, the integral is superficially divergent.

• $D=0$: Logarithmically divergent (e.g., $\int dk/k\sim \ln k$)
• $D=1$: Linearly divergent (e.g., $\int dk\sim k$)
• $D=2$: Quadratically divergent (e.g., $\int kdk\sim k^2$)

Let's see this applied to a specific case.

This power counting is "superficial" because sometimes symmetries (like gauge invariance) can cause cancellations that make the actual divergence less severe than what this simple counting predicts.

3. The Three Primitive Divergences of QED

Now we come to the central task of this lesson: identifying the fundamental one-loop diagrams in QED that are UV divergent. It turns out that in QED, all UV divergent diagrams are built from just a few "primitively divergent" building blocks. At the one-loop level, there are three of these.

This image shows the three key diagrams, which correct the basic components of our theory: the electron propagator, the photon propagator, and the interaction vertex.

Let's look at each one:

1. The Electron Self-Energy: This diagram represents a correction to the electron propagator. Physically, you can picture the electron emitting and then reabsorbing a virtual photon. This process modifies the electron's properties, specifically its mass.

2. The Vacuum Polarization: This is a correction to the photon propagator. A photon traveling through the vacuum can momentarily fluctuate into a virtual electron-positron pair, which then annihilates back into a photon. This cloud of virtual particles "polarizes" the vacuum, effectively screening the electric charge.

3. The Vertex Correction: This diagram modifies the fundamental QED interaction vertex. The incoming electron emits a virtual photon before interacting with the external photon, and then reabsorbs it. This corrects the coupling strength, $e$.

The following resource provides a systematic way to classify all potentially divergent diagrams in QED and confirms that only a finite number of types exist.

As the resource shows, all three of these primitive divergences are logarithmically divergent ($D=0$ for the vertex, but the self-energies are also log-divergent after accounting for symmetries). This is a crucial feature of a "renormalizable" theory. Theories with more severe divergences (e.g., $D>0$) are typically non-renormalizable, meaning that new, distinct infinities appear at every order of perturbation theory, rendering the theory non-predictive.

4. The Path Forward: Bare vs. Physical Quantities

We've established that our theory, when pushed beyond the simplest approximations, predicts infinite corrections to masses, charges, and interactions. This seems like a catastrophe.

The resolution, pioneered by Feynman, Schwinger, Tomonaga, and Dyson, is one of the deepest insights in modern physics. The parameters we write in our initial Lagrangian, like the mass $m_0$ and charge $e_0$, are not the physical quantities we measure in a laboratory. They are "bare" parameters. The physical mass $m$ and physical charge $e$ are the result of the bare parameter plus all its quantum corrections.

Physical Quantity (finite) = Bare Quantity + Infinite Loop Corrections

This equation can only hold if the bare quantities are themselves infinite, with a value carefully chosen to exactly cancel the infinities from the loop diagrams! This procedure is called renormalization. We absorb the infinities into a redefinition of our fundamental parameters.

To get a final conceptual summary of this idea, let's revisit the ZAP Physics video.

Conclusion

In this lesson, we have taken our first step beyond the world of tree-level diagrams and confronted the infinities that lie within quantum loops.

Key Takeaways:

• Higher-order corrections in QFT involve diagrams with closed loops.
• These loops contain unconstrained momenta, which must be integrated over, leading to ultraviolet (UV) divergences at high momenta.
• We can classify these divergences using the superficial degree of divergence (D).
• In QED, there are three fundamental one-loop UV divergent structures:
  1. Electron Self-Energy (corrects the electron propagator/mass)
  2. Vacuum Polarization (corrects the photon propagator/charge)
  3. Vertex Correction (corrects the interaction strength)
• The conceptual path to solving this problem is renormalization: absorbing these infinities into a redefinition of the "bare" parameters of the Lagrangian.

Preview of the Next Lesson:
We have identified the problem and sketched the conceptual solution. The next step is to develop a concrete mathematical toolkit to handle these infinite integrals. We cannot simply do algebra with the symbol $\infty$. We need a way to tame the divergence temporarily, perform our calculations, and only then see how the infinities cancel. This procedure is called regularization. In our next lesson, we will learn how to apply dimensional regularization to isolate UV divergences in one-loop integrals, the most widely used technique in modern QFT.