Part VI, Chapter 4

Renormalization Conditions

Fixing the finite parts: Connecting theory to experiment

4.1 The Ambiguity of Finite Parts

Regularization and counterterms remove divergences, but there's an ambiguity: we can addfinite pieces to counterterms without affecting UV finiteness!

$$\delta m^2 = \frac{C}{\epsilon} + (\text{finite part})$$

The pole 1/ε is fixed by demanding cancellation of divergences, but the finite part is arbitrary. Different choices define different renormalization schemes.

Renormalization Conditions:

To fix the finite parts, we impose renormalization conditions - physical requirements that determine the values of renormalized parameters m, λ, Z, etc.

These conditions define what we mean by the renormalized mass, coupling, etc.

4.2 On-Shell Renormalization Scheme

The most physically intuitive scheme: define renormalized parameters using physical, measurable quantities like particle masses and scattering amplitudes.

(a) Mass Renormalization Condition

The renormalized mass m is defined as the pole of the propagator:

$$\frac{i}{p^2 - m^2 - \Sigma(p^2)} \quad \xrightarrow{p^2 \to m^2} \quad \frac{i}{p^2 - m^2}$$

This requires the self-energy Σ(p²) to satisfy:

$$\Sigma(p^2 = m^2) = 0$$

💡Why the pole defines mass?

A free particle with mass m has propagator i/(p² - m²), which has a pole at p² = m². This pole corresponds to the particle being on-shell (satisfying E² = p² + m²).

When we include interactions, the propagator gets corrections Σ(p²). The physical massis still defined by where the pole occurs - the value of p² where the particle can propagate.

(b) Field Renormalization Condition

The field renormalization Z is fixed by requiring the residue of the pole to be 1:

$$\left.\frac{d\Sigma(p^2)}{dp^2}\right|_{p^2=m^2} = 0$$

Equivalently, the full propagator near the pole behaves like:

$$\frac{i}{p^2 - m^2 - \Sigma(p^2)} \approx \frac{iZ}{p^2 - m^2} \quad \text{with} \quad Z = 1$$

(c) Coupling Renormalization Condition

In φ⁴ theory, the coupling λ is defined by the 4-point scattering amplitude at a specific kinematic point (e.g., all particles at rest, s = 4m²):

$$\Gamma^{(4)}(s=4m^2, t=0, u=0) = -i\lambda$$

This fixes the finite part of δλ by measuring a physical scattering process!

On-Shell Conditions for φ⁴ Theory:

\begin{align*} \text{(1) Mass: } & \Sigma(p^2 = m^2) = 0 \\ \text{(2) Field: } & \frac{d\Sigma}{dp^2}\bigg|_{p^2=m^2} = 0 \\ \text{(3) Coupling: } & \Gamma^{(4)}(s=4m^2, t=u=0) = -i\lambda \end{align*}

4.3 On-Shell Scheme in QED

For QED, the on-shell conditions involve the electron mass and electric charge:

1. Electron Mass

The pole of the electron propagator defines the physical mass m_e:

$$\Sigma(\not{p} = m_e) = 0$$

2. Electric Charge

The charge e is defined by the Thomson scattering amplitude at q² = 0 (long-wavelength photon scattering off electron at rest):

$$\Gamma^\mu(q^2 = 0) = -ie\gamma^\mu$$

3. Photon Wave Function

The photon field renormalization is fixed by the transverse part of the photon propagator:

$$\left.\frac{d\Pi(q^2)}{dq^2}\right|_{q^2=0} = 0$$

Fine Structure Constant:

The famous α = e²/(4π) ≈ 1/137 is measured in the Thomson limit (q² → 0). At higher energies, α runs: α(M_Z) ≈ 1/128.

4.4 MS-bar Scheme

The modified minimal subtraction (MS-bar) scheme is purely mathematical:

MS-bar Prescription:

In dimensional regularization (d = 4 - ε), set counterterms to:

$$\delta X = -\frac{C_X}{\epsilon} - C_X(\ln 4\pi - \gamma_E) + 0$$

Subtract the pole, the ln(4π) - γ_E constants, and nothing else. No additional finite pieces.

In MS-bar, parameters like m(μ) and λ(μ) depend on the scale μ. They are not directly measurable but are convenient for calculations.

Example: MS-bar Mass

The MS-bar mass m̄(μ) is NOT the pole mass. They're related by:

$$m_{\text{pole}} = \bar{m}(\mu)\left[1 + \frac{\alpha_s(\mu)}{\pi} + O(\alpha_s^2)\right]$$

For the charm quark: m̄_c(m̄_c) = 1.27 GeV but m_pole ≈ 1.67 GeV (PDG).

4.5 Running Parameters

A key consequence: in MS-bar (and other mass-independent schemes), parameters run with scale:

$$m = m(\mu), \quad \lambda = \lambda(\mu), \quad \text{etc.}$$

Physical observables are independent of μ (renormalization group invariance), but parameters vary to compensate for changing μ.

💡Why do parameters run?

When we measure a coupling at energy scale E, we're probing physics at distances ~ 1/E. Virtual particles up to energy ~ E contribute to the effective coupling we measure.

At higher energies, more virtual modes contribute (we "see" more of the quantum cloud around particles), changing the effective coupling. This is the physics behind running!

4.6 Momentum Subtraction Schemes

Another approach: fix parameters at some reference momentum scale p² = -μ²:

\begin{align*} \Sigma(p^2 = -\mu^2) &= 0 \\ \Gamma^{(4)}(p_i^2 = -\mu^2) &= -i\lambda(\mu) \end{align*}

This is called the MOM scheme (momentum subtraction). It's useful for lattice QCD calculations where Euclidean momenta are natural.

4.7 Scheme Dependence and Independence

Scheme-Independent:

• Physical observables (cross sections, decay rates)
• Pole masses
• S-matrix elements
• Beta function (to leading order)

Scheme-Dependent:

• Renormalized mass m(μ)
• Renormalized coupling λ(μ)
• Z factors
• Finite parts of counterterms

Different schemes are related by finite renormalizations. Converting between schemes is straightforward once you understand the renormalization group (next chapter).

4.8 Comparison of Schemes

Common Renormalization Schemes

Each scheme has advantages for different applications

Aspect	Definition	Use Case
On-Shell	Pole at physical mass, residue = 1	Direct connection to experiment
MS-bar	Subtract 1/ε + ln(4π) - γ_E	Standard for running couplings, PDG
Momentum Subtraction	Fix value at specific momentum	Lattice QCD matching

Practical Usage:

On-shell: Best for low-energy precision tests (anomalous magnetic moment)
MS-bar: Standard for high-energy physics, running couplings, QCD
MOM: Useful for lattice QCD and non-perturbative methods

🎯 Key Takeaways

Counterterms have ambiguous finite parts - need renormalization conditions
On-shell scheme: parameters defined by physical masses and couplings
MS-bar scheme: minimal subtraction of 1/ε + ln(4π) - γ_E (standard today)
On-shell: m is pole mass; MS-bar: m̄(μ) is running mass
In MS-bar, parameters run with scale μ (but physics doesn't!)
Physical observables are scheme-independent
Different schemes related by finite renormalizations
Next: The renormalization group - how parameters flow with scale!

← Counterterms Next: Renormalization Group →

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