Quantum Field Theory Course

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Part IV, Chapter 3 | Page 3 of 3

The Ward Identity

How gauge invariance constrains quantum corrections and protects the photon mass

3.11 The Ward Identity for Scattering Amplitudes

The Ward identity is the quantum-mechanical consequence of gauge invariance. In its simplest form, it states that if $\mathcal{M}^\mu$ is any QED amplitude with an external photon carrying momentum $k$ and Lorentz index $\mu$, then:

$$\boxed{k_\mu \mathcal{M}^\mu = 0}$$

This means replacing the photon's polarization vector $\epsilon_\mu(k)$ by its momentum $k_\mu$ gives zero. Physically, this ensures that the longitudinal and timelike photon polarizations decouple from physical processes.

Derivation for a simple vertex. Consider Compton scattering, where an electron of momentum p absorbs a photon of momentum k and emits a photon. The two tree-level diagrams give:

$$\mathcal{M}^\mu = \bar{u}(p')\left[(-ie\gamma^\mu)\frac{i(\not{p}+\not{k}+m)}{(p+k)^2-m^2}(-ie\gamma^\nu) + (-ie\gamma^\nu)\frac{i(\not{p}-\not{k}'+m)}{(p-k')^2-m^2}(-ie\gamma^\mu)\right]u(p)\,\epsilon_\nu(k')$$

Contracting with $k_\mu$ and using the identity $\not{k} = (\not{p}+\not{k}-m) - (\not{p}-m)$:

$$k_\mu\gamma^\mu\frac{\not{p}+\not{k}+m}{(p+k)^2-m^2} = \frac{(\not{p}+\not{k}-m)(\not{p}+\not{k}+m)}{(p+k)^2-m^2} - \frac{(\not{p}-m)(\not{p}+\not{k}+m)}{(p+k)^2-m^2}$$

The first term simplifies using $(\not{p}+\not{k}-m)(\not{p}+\not{k}+m) = (p+k)^2 - m^2$, canceling the denominator. The second term vanishes when acting on $u(p)$ because $(\not{p}-m)u(p) = 0$ (Dirac equation). The same cancellation occurs in the crossed diagram, yielding $k_\mu\mathcal{M}^\mu = 0$.

3.12 The Ward-Takahashi Identity

The Ward identity for amplitudes is a special case of a more general identity for Green's functions, the Ward-Takahashi identity. It relates the vertex function $\Gamma^\mu(p',p)$ to the inverse electron propagator.

Define the exact (all-orders) electron propagator and vertex:

$$S_F^{-1}(p) = \not{p} - m - \Sigma(p), \qquad \Gamma^\mu(p',p) = \gamma^\mu + \Lambda^\mu(p',p)$$

where $\Sigma(p)$ is the electron self-energy and $\Lambda^\mu$ contains all vertex corrections. The Ward-Takahashi identity states:

$$\boxed{q_\mu\Gamma^\mu(p+q,p) = S_F^{-1}(p+q) - S_F^{-1}(p)}$$

where $q = p' - p$ is the photon momentum. This identity holds to all orders in perturbation theory. It is derived from the conservation of the electromagnetic current $\partial_\mu J^\mu = 0$ at the quantum level.

Derivation sketch. Start from the generating functional. The current conservation $\partial_\mu\langle J^\mu(x)\psi(y)\bar{\psi}(z)\rangle = 0$ (up to contact terms) gives, after Fourier transformation:

$$q_\mu\langle\tilde{J}^\mu(q)\tilde{\psi}(p+q)\tilde{\bar{\psi}}(-p)\rangle = \langle\tilde{\psi}(p)\tilde{\bar{\psi}}(-p)\rangle - \langle\tilde{\psi}(p+q)\tilde{\bar{\psi}}(-p-q)\rangle$$

Amputating the external legs (dividing by the exact propagators) and using the LSZ-like relation between the current insertion and the vertex function yields the Ward-Takahashi identity.

💡What the Ward-Takahashi Identity Means

The identity ties together two apparently different quantum corrections: the electron self-energy $\Sigma(p)$ and the vertex correction $\Lambda^\mu(p',p)$. It says that these corrections are not independent — they must be related in a specific way dictated by gauge invariance. Any approximation scheme that violates this relation is inconsistent with gauge symmetry and will produce unphysical results.

3.13 Consequences for Renormalization

The Ward-Takahashi identity has profound consequences for the renormalization of QED. The renormalized propagator and vertex involve renormalization constants:

$$\psi_{\text{bare}} = \sqrt{Z_2}\,\psi_R, \qquad A_{\text{bare}}^\mu = \sqrt{Z_3}\,A_R^\mu, \qquad e_{\text{bare}} = Z_e\,e_R$$

The key result is $Z_1 = Z_2$, where $Z_1$ renormalizes the vertex and $Z_2$renormalizes the electron field. To see this, evaluate the Ward-Takahashi identity at $q = 0$:

$$\Gamma^\mu(p,p) = \frac{\partial S_F^{-1}(p)}{\partial p_\mu}$$

Since $S_F^{-1}(p) = Z_2^{-1}(\not{p} - m_R - \Sigma_R(p))$, the derivative brings down a factor of $Z_2^{-1}$. Comparing with $\Gamma^\mu = Z_1^{-1}\gamma^\mu + \text{corrections}$, we deduce:

$$\boxed{Z_1 = Z_2}$$

This has a spectacular physical consequence. The renormalized charge is:

$$e_R = \frac{Z_2}{Z_1}\sqrt{Z_3}\,e_{\text{bare}} = \sqrt{Z_3}\,e_{\text{bare}}$$

The charge renormalization depends only on the photon field renormalization $Z_3$, not on any properties of the matter fields. This means all charged particles — electrons, muons, quarks — experience the same charge renormalization. The universality of electric charge is protected by gauge invariance.

3.14 Gauge Invariance at Loop Level

Consider the photon self-energy (vacuum polarization) $\Pi^{\mu\nu}(q)$, which receives contributions from virtual electron-positron loops. The Ward identity constrains its structure:

$$q_\mu\Pi^{\mu\nu}(q) = 0$$

This forces $\Pi^{\mu\nu}$ to be transverse:

$$\boxed{\Pi^{\mu\nu}(q) = (q^2 g^{\mu\nu} - q^\mu q^\nu)\,\Pi(q^2)}$$

The crucial point: there is no term proportional to $g^{\mu\nu}$ alone. A photon mass term would contribute $m_\gamma^2 g^{\mu\nu}$ to the self-energy. The Ward identity forbids this, so:

$$\Pi^{\mu\nu}(q)\Big|_{q=0} = 0 \qquad \Longrightarrow \qquad m_\gamma^2 = 0 \text{ to all orders}$$

The photon remains massless even after including all quantum corrections. This is a non-perturbative statement: gauge invariance protects the photon mass at every order of perturbation theory and beyond. Without this protection, radiative corrections would generically generate a photon mass of order the cutoff scale, destroying the long-range nature of electromagnetism.

💡Why Masslessness Needs Protection

In quantum field theory, if a symmetry does not forbid a parameter, loop corrections will generically generate it. A scalar field mass, for instance, receives quadratically divergent corrections ($\delta m^2 \sim \Lambda^2$). The photon mass would face the same fate — except gauge invariance provides an exact symmetry reason why $m_\gamma = 0$. The Ward identity is the mathematical implementation of this protection. This is the same mechanism that keeps gluons massless in QCD (SU(3) gauge invariance) and is the reason the Higgs mechanism is needed to give mass to the W and Z bosons in the electroweak theory.

3.15 Explicit Verification at One Loop

The one-loop vacuum polarization diagram gives:

$$i\Pi^{\mu\nu}(q) = (-1)(-ie)^2\int\frac{d^4\ell}{(2\pi)^4}\,\text{tr}\!\left[\gamma^\mu\frac{i(\not{\ell}+m)}{\ell^2-m^2+i\epsilon}\gamma^\nu\frac{i(\not{\ell}-\not{q}+m)}{(\ell-q)^2-m^2+i\epsilon}\right]$$

The factor of $(-1)$ is from the closed fermion loop. The trace is over spinor indices. Using dimensional regularization ($d = 4 - 2\varepsilon$) and Feynman parametrization, one obtains:

$$\Pi^{\mu\nu}(q) = (q^2g^{\mu\nu} - q^\mu q^\nu)\,\frac{-2\alpha}{\pi}\int_0^1 dx\,x(1-x)\left[\frac{2}{\varepsilon} - \ln\frac{m^2-x(1-x)q^2}{\mu^2} + \cdots\right]$$

The transverse structure $(q^2g^{\mu\nu} - q^\mu q^\nu)$ emerges automatically from the calculation when using a regularization that respects gauge invariance (such as dimensional regularization). This is the Ward identity in action at one loop: the gauge-violating term $g^{\mu\nu}$ cancels exactly.

Key Concepts

  • The Ward identity $k_\mu\mathcal{M}^\mu = 0$ ensures longitudinal photons decouple from physical amplitudes
  • The Ward-Takahashi identity $q_\mu\Gamma^\mu = S_F^{-1}(p+q) - S_F^{-1}(p)$ relates vertex and propagator corrections to all orders
  • Consequence: $Z_1 = Z_2$, so charge renormalization is universal ($e_R = \sqrt{Z_3}\,e_{\text{bare}}$)
  • The photon self-energy is forced to be transverse: $\Pi^{\mu\nu} = (q^2g^{\mu\nu} - q^\mu q^\nu)\Pi(q^2)$
  • The photon mass is protected: $m_\gamma = 0$ to all orders in perturbation theory
  • Dimensional regularization automatically preserves the Ward identity at loop level
  • The Ward identity is the quantum manifestation of classical gauge invariance
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