LSZ Reduction for Scalar Cross-Sections

Hello! Welcome to our next lesson.

In our previous session, we mastered Wick's theorem, a powerful tool for calculating time-ordered correlation functions like $⟨ Ω∣ T (ϕ (x_{1}) \dots ϕ (x_{n})) ∣Ω ⟩$ . We saw how this theorem breaks down complex operator products into sums of Feynman propagators, which we can visualize with diagrams.

This leaves us with a crucial question: how do these abstract correlation functions relate to the concrete, measurable quantities of a particle physics experiment, like scattering cross-sections? Today, we will build the two essential bridges that connect our theoretical framework to experimental reality.

This lesson covers the following learning outcome: Relate S-matrix elements to observable cross-sections using the LSZ reduction formula for scalar particles.

Specifically, we will:

Introduce the Lehmann-Symanzik-Zimmermann (LSZ) reduction formula, which forges the link between the correlation functions we calculate and the S-matrix elements that encode scattering amplitudes.
Derive the relationship between S-matrix elements and the differential cross-section, the primary observable in scattering experiments.

1. From Correlation Functions to the S-Matrix: The LSZ Reduction Formula

In a scattering experiment, we start with particles that are far apart and non-interacting in the distant past ( $t \to - \infty$ ), and we measure the resulting particles that are again far apart in the distant future ( $t \to + \infty$ ). These are called asymptotic states, or "in" and "out" states. The S-matrix is the operator whose elements, $⟨ out ∣ in ⟩$ , give the probability amplitude for a transition from a specific initial state to a specific final state.

Our goal is to relate these S-matrix elements to the correlation functions we've been calculating. The LSZ reduction formula accomplishes this.

The S matrix and the LSZ reduction formula

To understand the LSZ formula, we first need a more precise definition of these 'in' and 'out' states. The following reading introduces them as wavepackets created from the true interacting vacuum and develops the tools needed for the reduction formula.

Please read Section 2, 'Defining in- and out-states' (pages 3-6). Pay attention to how wavepackets are used to create localized, non-interacting asymptotic states and how the special time-dependent operators aψ(t) and aψ†(t) behave in the limits t → ±∞, allowing us to define the in- and out-states in equations (28a) and (28b).

The core idea behind the LSZ formula is to relate the creation and annihilation operators that build the 'in' and 'out' states, $a^{†} (\pm \infty)$ and $a (\pm \infty)$ , to the fundamental field operator $ϕ (x)$ . The key relation, which can be derived from the definitions you just read, is:

a_{p}^{†} (\infty) - a_{p}^{†} (- \infty) = - i \int d^{4} x e^{i p x} (□ + m^{2}) ϕ (x)

This allows us to systematically "reduce" an S-matrix element by replacing each particle's creation/annihilation operator with an integral over the field operator $ϕ (x)$ .

Let's see how this works in practice.

The S matrix and the LSZ reduction formula

The following section of the same document provides the main derivation of the LSZ reduction formula. It starts with an S-matrix element and applies the reduction trick one particle at a time.

Please read Section 3, 'LSZ reduction' (pages 7-9). Follow the steps from equation (32) to the final form of the formula in equation (41). The logic is to replace each a†(-∞) and a(∞) operator, which builds the 'in' and 'out' states, with an integral involving the Klein-Gordon operator acting on a field φ(x).

The final result, the LSZ Reduction Formula, for a scattering process with $m$ incoming particles (momenta $k_{i}$ ) and $n$ outgoing particles (momenta $p_{j}$ ) is:

⟨ p_{1} \dots p_{n} ∣ S ∣ k_{1} \dots k_{m} ⟩ = j = 1 \prod n (\frac{i}{Z} p_{j}^{2} \to m^{2} lim (p_{j}^{2} - m^{2}) \int d^{4} y_{j} e^{i p_{j} y_{j}}) i = 1 \prod m (\frac{i}{Z} k_{i}^{2} \to m^{2} lim (k_{i}^{2} - m^{2}) \int d^{4} x_{i} e^{- i k_{i} x_{i}}) ⟨ Ω∣ T (ϕ (y_{1}) \dots ϕ (x_{m})) ∣Ω ⟩

This formula looks complicated, but its physical and operational meaning is profound and surprisingly simple.

Feynman Diagram Expansion for a Vacuum Expectation Value in \phi^4 Theory — This image shows how a correlation function (top line) is expanded into a sum of Feynman diagrams. The LSZ formula connects the S-matrix element to the sum of all such diagrams. Notice the external lines corresponding to propagators for each particle. The LSZ procedure effectively 'amputates' these external lines.

Let's break it down:

Correlation Function: The formula starts with the Fourier transform of the $(n + m)$ -point time-ordered correlation function. This is exactly what we learned to compute using Wick's theorem.
Poles: The Fourier transform of a correlation function has poles at $p^{2} = m^{2}$ for each external particle. These poles come from the propagators of the external legs of the Feynman diagrams.
Amputation: The factors of $(p_{j}^{2} - m^{2})$ and $(k_{i}^{2} - m^{2})$ in the LSZ formula are designed to precisely cancel these poles. This mathematical procedure is called amputation—it's like cutting off the external legs of the Feynman diagrams, leaving only the central, interacting part.
On-Shell Limit: The limits $p_{j}^{2} \to m^{2}$ and $k_{i}^{2} \to m^{2}$ enforce that the external particles are real, physical particles satisfying the relativistic energy-momentum relation.
Wavefunction Renormalization (Z): The factor $Z$ accounts for the fact that the interaction modifies the relationship between the field operator $ϕ (x)$ and the single-particle states. For the tree-level calculations we'll be doing soon, we can approximate $Z = 1$ .

In short, the LSZ formula tells us that to get the scattering amplitude, we must compute the corresponding Green's function, Fourier transform it, and then amputate the external legs.

2. From the S-Matrix to Cross-Sections

We now have the S-matrix element $⟨ f ∣ S ∣ i ⟩$ , which is a complex number representing the amplitude for the process $i \to f$ . The next step is to relate this to a physical observable.

First, we separate the part of the S-matrix that corresponds to no interaction. We define the T-matrix (or transfer matrix) for this purpose:

S = 1 + i T

Here, $1$ represents the case where particles don't scatter, and $i T$ contains all the interesting dynamics.

Due to overall energy-momentum conservation, the matrix elements of $T$ must contain a delta function. We factor this out to define the invariant matrix element, $M$ :

⟨ f ∣ i T ∣ i ⟩ = i M_{f i} (2 π)^{4} δ^{(4)} (\sum p_{f} - \sum p_{i})

The invariant matrix element $M$ is the Lorentz-invariant quantity that contains the core dynamics of the scattering process. Our goal in any scattering calculation is to compute $M$ .

Now, how do we get from $M$ to a cross-section? The cross-section, $σ$ , represents the effective area that the target particle presents to the projectile for a scattering event to occur. It relates the number of scattering events to the incident flux of particles.

Quantum Field Theory I - Lecture 11

The following video provides a detailed, first-principles derivation of the general formula for the scattering cross-section. It starts from the basic definition of probability and carefully integrates over all possibilities to arrive at the final, Lorentz-invariant expression. The level of detail in this derivation should align well with your background.

This is a long and detailed derivation. I recommend breaking it into three parts: Setup (5:39 - 11:21): Watch how the probability for a scattering event is formulated in terms of the S-matrix element and wavepackets for incoming particles. The General Formula (38:58 - 55:57): This part presents the final, general formula for the differential cross-section, which is the main result. It also introduces the crucial concepts of the Lorentz-invariant flux factor and the n-body phase space. The video segment from 11:21 to 38:58 contains the extremely detailed algebraic steps to get here, which you can skim or watch later if you're curious about every single delta-function integration. Special Case: 2→2 Scattering (55:57 - 1:12:50): See how the general formula simplifies for the very common case of 2-particle to 2-particle scattering in the center-of-mass frame. This result will be used frequently in upcoming lessons.

Let's summarize the essential formulas from that derivation.

The differential cross-section for a process $A + B \to 1 + 2 + \dots + n$ is given by:

d σ = \frac{1}{4 ( p _{A} \cdot p _{B} ) ^{2} - m _{A}^{2} m _{B}^{2}} \times ∣ M ∣^{2} \times d Π_{n} \times S

Let's break this down:

$\frac{1}{4 ( p _{A} \cdot p _{B} ) ^{2} - m _{A}^{2} m _{B}^{2}}$ : This is the inverse of the incident flux factor. It is Lorentz invariant and in the center-of-mass frame simplifies to $1/ (4 p_{C M} E_{C M})$ , where $p_{C M}$ is the magnitude of the incoming 3-momentum and $E_{C M}$ is the total energy.
$∣ M ∣^{2}$ : The squared invariant matrix element, which contains all the dynamics of the interaction. This is what we calculate using Feynman diagrams.
$d Π_{n}$ : The element of Lorentz-invariant phase space (LIPS) for the $n$ final-state particles. It represents the number of available final states. $d Π_{n} = (2 π)^{4} δ^{(4)} P_{in} - f = 1 \sum n p_{f} f = 1 \prod n \frac{d ^{3} p _{f}}{( 2 π ) ^{3} 2 E _{f}}$
$S$ : A statistical factor of $1/ k!$ for each group of $k$ identical particles in the final state.

For the common case of 2 $\to$ 2 scattering ( $A + B \to C + D$ ) in the center-of-mass frame, this formula simplifies to the differential cross-section with respect to the solid angle $Ω$ :

\frac{d σ}{d Ω} = \frac{1}{64 π ^{2} E _{C M}^{2}} \frac{∣ p _{f} ∣}{∣ p _{i} ∣} ∣ M ∣^{2}

where $E_{C M}$ is the total center-of-mass energy, $∣ p_{i} ∣$ is the magnitude of the initial 3-momentum of either incoming particle, and $∣ p_{f} ∣$ is the magnitude of the final 3-momentum of either outgoing particle.

Test your understanding!

Consider elastic scattering of two identical particles, $A + A \to A + A$ , where each particle has mass $m$ . How does the formula for $d σ / d Ω$ in the center-of-mass frame simplify? What is the statistical factor $S$ in this case?

Show answer

For elastic scattering, energy is conserved, so the final momentum magnitude is equal to the initial momentum magnitude: $∣ p_{f} ∣ = ∣ p_{i} ∣$ . The formula simplifies to:

\frac{d σ}{d Ω} = \frac{1}{64 π ^{2} E _{C M}^{2}} ∣ M ∣^{2}

Since there are two identical particles in the final state, the statistical factor is $S = 1/2!$ . To get the total cross-section $σ$ , you would integrate $d σ / d Ω$ over the solid angle. However, because the final particles are identical, a scattering event where particle 1 goes into angle $θ$ and particle 2 into $π - θ$ is indistinguishable from the reverse. To avoid double-counting, we either integrate over only half the solid angle (e.g., $0 \leq θ \leq π /2$ ) or we integrate over the full solid angle and multiply by $S = 1/2$ .

Conclusion

Today we have forged the final links in the chain connecting the abstract formalism of QFT to physical reality. We now have a complete pipeline for predicting the outcome of a scattering experiment.

Key Takeaways:

The LSZ Reduction Formula is the dictionary that translates between time-ordered correlation functions and S-matrix elements.
Operationally, the LSZ formula instructs us to calculate the appropriate correlation function, Fourier transform it, and "amputate" the external propagators by canceling their poles.
The S-matrix is related to the invariant matrix element $M$ , which contains the Lorentz-invariant dynamics of the interaction.
The physically observable cross-section, $σ$ , is calculated by squaring $M$ and integrating over the available final-state phase space, corrected for the incident flux.

Preview of the Next Lesson:

We have established the full procedure: use Wick's theorem to find the correlation function, use LSZ to get $M$ , and then integrate $∣ M ∣^{2}$ to get $σ$ . This is still a cumbersome process. In our next lesson, we will use the LSZ formula as our foundation to derive the famous Feynman rules. These rules are a powerful shortcut that will allow us to bypass the formal machinery of Wick's theorem and LSZ, enabling us to write down the expression for the invariant matrix element $M$ directly from a set of simple diagrams.

Recommended resources

1. From Correlation Functions to the S-Matrix: The LSZ Reduction Formula

2. From the S-Matrix to Cross-Sections

Conclusion