Quantum Field Theory Course

Total: 440+ pages covering classical field theory through the Standard Model

Part III, Chapter 4 — Page 2 of 3

The LSZ Reduction Formula

Connecting time-ordered correlation functions to physical scattering amplitudes

4.7 From Green Functions to the S-Matrix

We know how to compute correlation functions $G^{(n)}(x_1,\ldots,x_n)$ using the path integral and Feynman diagrams. But experiments measure scattering cross sections, which are determined by S-matrix elements $\langle f|S|i\rangle$. The LSZ reduction formula (Lehmann, Symanzik, Zimmermann, 1955) provides the bridge.

The key question is: how do we extract the probability amplitude for particles with definite momenta from correlation functions defined in position space?

💡The Physical Picture

In a scattering experiment, particles start far apart (asymptotic past), interact briefly, and end far apart (asymptotic future). The LSZ formula tells us exactly how to project the correlation function onto these asymptotic single-particle states, connecting the abstract n-point function to the measurable scattering amplitude.

4.8 Asymptotic States and Field Renormalization

In the interacting theory, the field $\phi(x)$ does not simply create single-particle states. We must carefully define asymptotic states. The key assumption is that at $t \to \pm\infty$, the interacting field behaves like a free field up to a normalization:

$$\langle \Omega|\phi(x)|\mathbf{p}\rangle = \sqrt{Z}\, e^{-ip\cdot x}$$

where $|\Omega\rangle$ is the interacting vacuum, $|\mathbf{p}\rangle$ is a single-particle state with momentum $\mathbf{p}$, and $Z$ is the wavefunction renormalization constant (also called field strength renormalization).

The constant $Z$ satisfies $0 < Z \leq 1$ and accounts for the fact that $\phi(x)$ in the interacting theory can create multi-particle states as well. In the free theory, $Z = 1$.

We define the asymptotic fields that do create normalised single-particle states:

$$\phi(x) \xrightarrow{t\to-\infty} \sqrt{Z}\,\phi_{\text{in}}(x), \qquad \phi(x) \xrightarrow{t\to+\infty} \sqrt{Z}\,\phi_{\text{out}}(x)$$

where $\phi_{\text{in}}$ and $\phi_{\text{out}}$ are free fields satisfying $(\Box + m^2)\phi_{\text{in/out}} = 0$ with $m$ the physical (renormalized) mass.

4.9 The LSZ Reduction Formula

Consider the scattering process $p_1,\ldots,p_m \to q_1,\ldots,q_n$. The LSZ formula states:

$$\boxed{\langle q_1\cdots q_n|S|p_1\cdots p_m\rangle = \left(\frac{i}{\sqrt{Z}}\right)^{m+n}\prod_{i=1}^{m}\int d^4x_i\,e^{ip_i\cdot x_i}(\Box_{x_i}+m^2)\prod_{j=1}^{n}\int d^4y_j\,e^{-iq_j\cdot y_j}(\Box_{y_j}+m^2)\,G^{(m+n)}(x_1,\ldots,x_m,y_1,\ldots,y_n)}$$

Each external particle contributes:

  • An integral $\int d^4x\, e^{\pm ip\cdot x}$ — Fourier transform to momentum space
  • A Klein-Gordon operator $(\Box + m^2)$ — projects onto the on-shell single-particle state
  • A factor $i/\sqrt{Z}$ — accounts for field renormalization

💡What Does the KG Operator Do?

The Klein-Gordon operator $(\Box + m^2)$ is the inverse of the propagator. When acting on a Green function with an external propagator $i/(p^2-m^2)$, it amputates that leg, removing the external propagator. The result is finite only if the external momentum is on-shell ($p^2 = m^2$), where the propagator has a pole. LSZ extracts the residue of this pole, which is the physical scattering amplitude.

4.10 Proof Sketch of the LSZ Formula

The proof proceeds in several steps. We outline the key ideas for a single incoming particle (the generalization is straightforward).

Step 1: Create an Incoming Particle

We use the creation operator for the asymptotic "in" field:

$$a_{\text{in}}^\dagger(\mathbf{p}) = -i\int d^3x\, e^{-ip\cdot x}\overset{\leftrightarrow}{\partial_0}\phi_{\text{in}}(x)$$

where $f\overset{\leftrightarrow}{\partial_0}g = f(\partial_0 g) - (\partial_0 f)g$. Since $\phi(x) \to \sqrt{Z}\phi_{\text{in}}(x)$ as $t\to -\infty$:

$$\sqrt{Z}\,a_{\text{in}}^\dagger(\mathbf{p}) = -i\lim_{t\to-\infty}\int d^3x\, e^{-ip\cdot x}\overset{\leftrightarrow}{\partial_0}\phi(x)$$

Step 2: Convert Time Limits to Spacetime Integrals

The difference between the $t\to+\infty$ and $t\to-\infty$ limits can be written as a total time derivative:

$$\sqrt{Z}\left(a_{\text{out}}^\dagger(\mathbf{p}) - a_{\text{in}}^\dagger(\mathbf{p})\right) = -i\int d^4x\,\partial_0\left[e^{-ip\cdot x}\overset{\leftrightarrow}{\partial_0}\phi(x)\right]$$

Evaluating the time derivative and using $p^2 = m^2$ (on-shell condition) to simplify $\partial_0^2 e^{-ip\cdot x}$:

$$\sqrt{Z}\left(a_{\text{out}}^\dagger - a_{\text{in}}^\dagger\right) = -i\int d^4x\, e^{-ip\cdot x}(\Box_x + m^2)\phi(x)$$

Step 3: Build Up the S-Matrix

Applying this procedure to each external particle — incoming particles via $a_{\text{in}}^\dagger$ and outgoing particles via $a_{\text{out}}$ — we insert the interacting field operators between the vacuum. The time-ordering emerges naturally because incoming particles are created in the far past and outgoing particles are annihilated in the far future. After $m+n$ such reductions, we arrive at the full LSZ formula.

4.11 Amputated Diagrams and On-Shell Conditions

In momentum space, the LSZ formula takes a particularly clean form. The Fourier transform of $G^{(n)}$ near the mass shell has poles from external propagators:

$$\tilde{G}^{(n)}(p_1,\ldots,p_n) \xrightarrow{p_i^2 \to m^2} \prod_{i=1}^{n}\frac{iZ}{p_i^2 - m^2 + i\epsilon} \cdot \mathcal{M}(p_1,\ldots,p_n) \cdot (2\pi)^4\delta^4\!\left(\sum p_i\right)$$

The amputated Green function is obtained by removing the external propagators:

$$\boxed{G^{(n)}_{\text{amp}}(p_1,\ldots,p_n) = \prod_{i=1}^{n}\left[\frac{p_i^2 - m^2}{i}\right]\tilde{G}^{(n)}(p_1,\ldots,p_n)}$$

The scattering amplitude is then:

$$\boxed{i\mathcal{M}(p_1,\ldots,p_n) = Z^{-n/2}\,G^{(n)}_{\text{amp}}(p_1,\ldots,p_n)\Big|_{p_i^2 = m^2}}$$

Practical Recipe

  1. Draw all Feynman diagrams for $G^{(n)}$ to the desired order
  2. Amputate: Remove the external propagators from each diagram
  3. On-shell: Set $p_i^2 = m^2$ for all external momenta
  4. Multiply by $Z^{-n/2}$ (often $Z=1$ at leading order)
  5. The result is the invariant amplitude $i\mathcal{M}$

💡Why Amputation?

External propagators represent the propagation of the asymptotic free particles from/to infinity. Since the detector measures only on-shell particles, the external propagators carry no physical information about the scattering interaction itself. The amputated diagram isolates the interaction kernel — the part that actually describes what happens during the collision.

Key Concepts — LSZ Reduction

  • LSZ connects correlation functions to S-matrix elements (measurable scattering)
  • Asymptotic states: $\phi \to \sqrt{Z}\phi_{\text{in/out}}$ at $t \to \mp\infty$
  • Each external leg contributes $(i/\sqrt{Z})\int d^4x\, e^{\pm ip\cdot x}(\Box + m^2)$
  • The Klein-Gordon operator amputates external propagators
  • On-shell condition $p^2 = m^2$ selects single-particle poles
  • Scattering amplitude = amputated, on-shell Green function times $Z^{-n/2}$
  • At tree level, $Z = 1$ and amputation is the only step needed
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