Part VI, Chapter 3

Counterterms

Absorbing infinities into parameter redefinitions

3.1 Bare vs. Renormalized Parameters

The key insight of renormalization: the parameters appearing in the Lagrangian are not physical. They are called bare parameters (m₀, λ₀, etc.) and are infinite in the continuum limit.

The Renormalization Idea:

Split bare parameters into renormalized (physical) parts plus counterterms:

\begin{align*} m_0^2 &= m^2 + \delta m^2 \\ \lambda_0 &= \lambda + \delta\lambda \\ \phi_0 &= \sqrt{Z}\phi \end{align*}

The counterterms δm², δλ, δZ are infinite but chosen to cancel loop divergences exactly!

💡Why are bare parameters infinite?

Imagine the electron. The "bare" electron (without any QED interactions) has some mass m₀. But in reality, the electron constantly emits and reabsorbs virtual photons, which contribute infinite corrections to its mass.

The physical mass m we measure in experiments is m₀ + (infinite quantum corrections). For this sum to be finite, m₀ itself must be infinite in a precisely tuned way!

This seems crazy, but it works: all physical predictions depend only on m, not on m₀.

3.2 Example: φ⁴ Theory

Consider the scalar field theory with Lagrangian:

$$\mathcal{L} = \frac{1}{2}(\partial_\mu\phi_0)^2 - \frac{1}{2}m_0^2\phi_0^2 - \frac{\lambda_0}{4!}\phi_0^4$$

Substitute the renormalized field and parameters:

\begin{align*} \phi_0 &= \sqrt{Z}\phi \\ m_0^2 &= m^2 + \delta m^2 \\ \lambda_0 &= \lambda + \delta\lambda \end{align*}

The Lagrangian becomes:

$$\mathcal{L} = \mathcal{L}_{\text{ren}} + \mathcal{L}_{\text{ct}}$$

where the renormalized Lagrangian is:

$$\mathcal{L}_{\text{ren}} = \frac{1}{2}(\partial_\mu\phi)^2 - \frac{1}{2}m^2\phi^2 - \frac{\lambda}{4!}\phi^4$$

and the counterterm Lagrangian is:

$$\mathcal{L}_{\text{ct}} = \frac{\delta Z}{2}(\partial_\mu\phi)^2 - \frac{\delta m^2}{2}\phi^2 - \frac{\delta\lambda}{4!}\phi^4$$

where δZ = Z - 1.

Feynman Rules for Counterterms:

Counterterms contribute new vertices in Feynman diagrams:

Field renormalization: × denoting δZ correction to propagator
Mass counterterm: Blob on propagator = -iδm²
Coupling counterterm: Special 4-vertex = -iδλ

3.3 One-Loop Renormalization of φ⁴

Let's work out the counterterms needed at one-loop order in φ⁴ theory using dimensional regularization (d = 4 - ε).

(a) Tadpole Diagram

The one-loop tadpole gives:

$$T = -\frac{i\lambda}{2} \int \frac{d^dk}{(2\pi)^d} \frac{1}{k^2 - m^2} = -\frac{\lambda m^2}{32\pi^2}\left(\frac{2}{\epsilon} - \ln\frac{m^2}{\mu^2} + \text{finite}\right)$$

This contributes to the 1-point function. To cancel it, we need:

$$\delta m^2 = \frac{\lambda m^2}{32\pi^2\epsilon} + \text{finite}$$

(b) Self-Energy

The one-loop self-energy correction (loop with two external legs):

$$\Sigma(p^2) = \frac{\lambda}{2} \int \frac{d^dk}{(2\pi)^d} \frac{1}{k^2 - m^2} = \frac{\lambda}{32\pi^2}\left(\frac{2}{\epsilon} - \ln\frac{m^2}{\mu^2} + \text{finite}\right)$$

This modifies the propagator. At p² = m² (on-shell), we need counterterms to maintain the correct pole structure.

(c) Vertex Correction

The one-loop correction to the 4-point vertex involves three diagrams (s-channel, t-channel, u-channel):

$$\Gamma^{(4)} = -i\lambda - i\frac{3\lambda^2}{32\pi^2}\left(\frac{2}{\epsilon} - \ln\frac{m^2}{\mu^2} + \text{finite}\right) + \cdots$$

The coupling counterterm is:

$$\delta\lambda = \frac{3\lambda^2}{16\pi^2\epsilon} + \text{finite}$$

Pattern at One-Loop:

All counterterms have the structure:

$$\delta X = \frac{C_X}{\epsilon} + \text{finite}$$

The pole part (1/ε) is universal and scheme-independent. The finite part depends on the renormalization scheme (MS, MS-bar, on-shell, etc.).

3.4 Counterterms in QED

In QED, the bare Lagrangian is:

$$\mathcal{L}_0 = -\frac{1}{4}F_{\mu\nu}^0 F_0^{\mu\nu} + \bar{\psi}_0(i\gamma^\mu\partial_\mu - m_0)\psi_0 - e_0\bar{\psi}_0\gamma^\mu\psi_0 A_\mu^0$$

We introduce renormalization constants:

\begin{align*} A_\mu^0 &= \sqrt{Z_3} A_\mu \\ \psi_0 &= \sqrt{Z_2} \psi \\ m_0 &= Z_m m \\ e_0 &= Z_e e \end{align*}

The counterterm Lagrangian becomes:

\begin{align*} \mathcal{L}_{\text{ct}} &= -\frac{\delta Z_3}{4}F_{\mu\nu}F^{\mu\nu} + \delta Z_2 \bar{\psi}i\gamma^\mu\partial_\mu\psi \\ &\quad - \delta m \bar{\psi}\psi - \delta e \bar{\psi}\gamma^\mu\psi A_\mu \end{align*}

Three Key Counterterms in QED:

1. Photon Field Renormalization (δZ₃):

From vacuum polarization (electron loop). Modifies photon propagator and running of α.

$$\delta Z_3 = -\frac{e^2}{12\pi^2\epsilon} + \text{finite}$$

2. Electron Field Renormalization (δZ₂):

From electron self-energy (photon loop). Affects wave function normalization.

$$\delta Z_2 = -\frac{e^2}{8\pi^2\epsilon} + \text{finite}$$

3. Mass Counterterm (δm):

Logarithmically divergent mass shift from virtual photons.

$$\delta m = -\frac{3e^2 m}{16\pi^2\epsilon} + \text{finite}$$

3.5 Ward Identity and Charge Renormalization

A remarkable fact in QED: gauge invariance (via the Ward-Takahashi identity) implies:

$$Z_e = \frac{Z_2}{\sqrt{Z_3}}$$

This means the charge counterterm is not independent:

$$\delta Z_e = \delta Z_2 - \frac{1}{2}\delta Z_3$$

💡Physical Meaning of Ward Identity

The Ward identity is a consequence of gauge invariance (U(1) symmetry in QED). It ensures that the electric charge is not renormalized in the same way as other quantities.

More precisely: while Z₂ and Z₃ are individually infinite, their ratio Z₂/√Z₃ remains finite. This protects gauge coupling from certain divergences and is crucial for renormalizability.

3.6 Systematic Renormalization: BPHZ

For multi-loop diagrams with nested subdivergences, we need a systematic procedure. TheBogoliubov-Parasiuk-Hepp-Zimmermann (BPHZ) method provides this:

BPHZ Algorithm:

Identify all divergent subdiagrams (not just superficial D ≥ 0)
Order them from innermost to outermost
Apply counterterms to each subdiagram recursively
The R-operation: subtract divergent parts at each level

The BPHZ prescription ensures that all divergences are removed to all orders in perturbation theory for renormalizable theories.

Example: Two-Loop Vertex

A two-loop vertex correction contains one-loop subdivergences. We must:

Renormalize inner loops first (add their counterterms)
Then renormalize the remaining overall divergence
This recursive procedure gives finite results

3.7 Physical Interpretation

Field Renormalization (Z factors)

Z ≠ 1 means the wave function normalization receives quantum corrections. The bare field φ₀ and physical field φ differ by √Z. This affects how we normalize external states in scattering amplitudes.

Mass Renormalization (δm²)

Virtual particle loops contribute to the particle's self-energy, shifting its mass. Thephysical mass m is what we measure (pole of propagator), not the bare mass m₀.

Coupling Renormalization (δλ, δe)

Vertex corrections modify interaction strength. The effective coupling depends on the energy scale - this is the physics of running couplings, which we'll study in detail in Chapter 6.

🎯 Key Takeaways

Bare parameters (m₀, λ₀) in Lagrangian are infinite and unphysical
Split: bare = renormalized + counterterm (m₀² = m² + δm²)
Counterterm Lagrangian ℒ_ct has same form as original theory
Counterterms δm², δλ, δZ are chosen to cancel loop divergences
In dim reg: counterterms ~ 1/ε (pole in d = 4 - ε)
Ward identity in QED: Z_e = Z₂/√Z₃ (gauge invariance!)
BPHZ prescription handles subdivergences systematically
Next: Fixing finite parts via renormalization conditions!

← Regularization Schemes Next: Renormalization Conditions →

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