On-Shell Renormalization: Counteracting UV Divergences

1. The Big Picture: Bare vs. Physical Quantities

The core idea of renormalization is to distinguish between the parameters that appear in our initial Lagrangian and the quantities we actually measure in a laboratory.

To build intuition, let's start with a conceptual overview.

This idea is the foundation of renormalization. The parameters in our original QED Lagrangian—the mass $m_0$, the charge $e_0$, and the fields ${\psi }_0,A_{0\mu }$—are bare quantities. They are theoretical constructs that cannot be directly measured because they don't include the effects of quantum fluctuations.

The quantities we measure—the electron's mass $m$, its charge $e$, and the fields that create physical particles—are the renormalized or physical quantities. Renormalization is the dictionary that translates between these two descriptions.

2. The Counterterm Method

The standard procedure for implementing this translation is the counterterm method. The strategy is to rewrite the theory entirely in terms of the physical quantities we want to work with, while systematically hiding the infinities inside new interaction terms called "counterterms".

Here's the process:

1. Start with the Bare Lagrangian: We begin with the QED Lagrangian written in terms of bare quantities.

   $${\mathcal{L}}_{\text{bare}}=-\frac{1}{4}(F_{0\mu \nu }{)}^2+{\bar{\psi }}_0(i/\partial -m_0){\psi }_0-e_0{\bar{\psi }}_0{\gamma }^{\mu }{\psi }_0{A}_{0\mu }$$

2. Relate Bare and Renormalized Quantities: We define the bare quantities in terms of the physical (renormalized) ones using renormalization constants $Z_i$.

   • Bare field ${\psi }_0=Z_2^{1/2}\psi$
   • Bare field $A_{0\mu }=Z_3^{1/2}{A}_{\mu }$
   • Bare mass and charge are related via $m_0=\frac{m+{\delta }_m}{Z_2}$ and $e_0=\frac{Z_1}{Z_2{Z}_3^{1/2}}e$.

   The $Z_i$ factors are formally divergent series in the coupling constant $\alpha$, starting with $Z_i=1+\mathcal{O}(\alpha )$.

3. Split the Lagrangian: Substitute these relations back into ${\mathcal{L}}_{\text{bare}}$ and rearrange it into two parts: ${\mathcal{L}}_{\text{bare}}={\mathcal{L}}_{\text{physical}}+{\mathcal{L}}_{\text{counterterm}}$.

   • ${\mathcal{L}}_{\text{physical}}$ has the same form as our original Lagrangian, but written with the physical fields and parameters. This part defines the propagators and vertices for our perturbative calculations.
   • ${\mathcal{L}}_{\text{counterterm}}$ contains all the leftover pieces, which are expressed in terms of counterterms like ${\delta }_1=Z_1-1$, ${\delta }_2=Z_2-1$, ${\delta }_3=Z_3-1$, and ${\delta }_m$.

The following reading details this procedure, giving the explicit form of the QED counterterm Lagrangian and its corresponding Feynman rules.

By treating the counterterm vertices as new interactions, any loop diagram that gives a UV divergence will now be accompanied by a counterterm diagram of the same order. The magic of renormalization is that the counterterms will be chosen to precisely cancel these divergences.

3. The On-Shell Renormalization Scheme

How do we fix the values of the counterterms ${\delta }_1,{\delta }_2,{\delta }_3,{\delta }_m$? We need a set of physical conditions. The on-shell scheme provides these conditions by demanding that our renormalized parameters correspond to the properties of real, physical particles.

Your background in physics provides a strong intuition for what defines a particle's mass and charge. A particle's mass is its rest energy, measured when it's on its mass shell ($p^2=m^2$). Its charge is what we measure in the low-energy limit (e.g., through Thomson scattering). The on-shell scheme elevates these physical definitions into mathematical constraints on our theory.

The Lehmann-Symanzik-Zimmermann (LSZ) reduction formula, which you may have encountered, provides the rigorous link. It states that S-matrix elements for scattering processes are determined by the poles and residues of the full (or "dressed") Green's functions. This motivates the on-shell conditions:

1. Electron Mass: The full electron propagator should have a pole at $/p=m$, where $m$ is the physical electron mass.
2. Photon Mass: The full photon propagator should have a pole at $k^2=0$, as the photon is massless.
3. Field Normalization: To ensure our fields create single-particle states with unit probability, the residue of the pole in the respective propagators must be 1 (or $i$ by convention).
4. Electric Charge: The physical charge $e$ is defined as the electron-photon coupling strength in the zero-momentum-transfer limit ($q\to 0$), corresponding to a classical measurement.

These physical requirements can be translated into precise mathematical conditions on the one-particle-irreducible (1PI) functions we've been calculating.

Let's rephrase the on-shell conditions in terms of these 1PI functions.

To summarize and formalize the conditions from the reading:

• Electron Propagator: The dressed propagator is $\frac{i}{/p-m-{\Sigma }_{\text{net}}(/p)}$, where ${\Sigma }_{\text{net}}={\Sigma }_{\text{loop}}+{\Sigma }_{\text{ct}}$. The conditions that the pole is at $/p=m$ with residue $i$ translate to:

  1. ${\Sigma }_{\text{net}}(/p=m)=0$
  2. $\frac{d{\Sigma }_{\text{net}}}{d/p}{|}_{/p=m}=0$

• Photon Propagator: The dressed propagator is $\frac{-i(g^{\mu \nu }-\dots )}{k^2(1-{\Pi }_{\text{net}}(k^2))}$. The condition that the pole is at $k^2=0$ with the correct residue is:
  3. ${\Pi }_{\text{net}}(k^2=0)=0$

• Vertex Function (Charge): The condition that the coupling is exactly $e$ at zero momentum transfer means the loop corrections to the vertex must vanish in that limit. Denoting the 1PI vertex as ${\Gamma }^{\mu }={\gamma }^{\mu }+\delta {\Gamma }_{\text{net}}^{\mu }$, we require:
  4. ${\lim }_{q\to 0}\bar{u}(p)[\delta {\Gamma }_{\text{net}}^{\mu }]u(p)=0$ (with electrons on-shell)

These four conditions are sufficient to uniquely determine the four counterterms ${\delta }_m,{\delta }_2,{\delta }_3,{\delta }_1$ order by order in perturbation theory.

4. An Example: Renormalizing the Photon Propagator

Let's see this in action for the vacuum polarization. The net 1PI photon self-energy is the sum of the loop diagram and the counterterm diagram:

From our previous lessons and the resources, we know the structure of these terms:

The loop function ${\Pi }_{\text{loop}}(k^2)$ contains a UV divergence. The net function is thus:

Now, we apply our on-shell renormalization condition: ${\Pi }_{\text{net}}(k^2=0)=0$. This forces our choice of the counterterm ${\delta }_3$:

This is a profound result. We have defined the counterterm ${\delta }_3$ to be equal to the value of the divergent loop function at $k^2=0$. This means ${\delta }_3$ is itself divergent, precisely tuned to cancel the divergence in the loop calculation.

The final, physical, renormalized photon self-energy is:

This new function $\hat{\Pi }(k^2)$ is manifestly zero at $k^2=0$ and, crucially, it is UV finite. This is the function that gives rise to the physical phenomenon of the running of the fine-structure constant. The same procedure applies to the electron self-energy and vertex function, using the other renormalization conditions to fix ${\delta }_m,{\delta }_2$, and ${\delta }_1$.

Conclusion

In this lesson, we have constructed the formal machinery of renormalization in QED. We have moved from simply identifying divergences to systematically taming them.

Key Takeaways:

• Renormalization is a procedure for relating unobservable bare parameters in a Lagrangian to physical, measurable quantities.
• The counterterm method achieves this by splitting the Lagrangian into a physical part and a counterterm part. The counterterms are treated as new interaction vertices.
• The on-shell renormalization scheme defines the values of the counterterms by imposing conditions on the poles and residues of the full propagators, ensuring they describe particles with the correct physical mass and charge.
• By choosing the (divergent) counterterms appropriately, the sum of loop diagrams and counterterm diagrams for any physical process becomes UV finite, order by order in perturbation theory.

Preview of the Next Lesson:
Throughout this process, we've relied on the fact that the divergences have a specific structure. For example, the photon self-energy divergence did not generate a mass term, which would have been a disaster. Why is this? This is not an accident; it is a consequence of the underlying gauge symmetry of QED. In our next lesson, we will formally introduce the Ward-Takahashi identity, a powerful mathematical expression of this symmetry. We will see how it guarantees relations between the renormalization constants (specifically, $Z_1=Z_2$) and protects the photon's masslessness, ultimately ensuring that QED is a consistent, renormalizable theory.