From Scattering to LSZ

A scattering calculation needs more than a propagator. It needs the interaction connecting the selected external particles, the conversion from field insertions to normalized states, and the flux and final-state counting appropriate to the measurement. LSZ supplies the middle step under stable-particle and asymptotic-state assumptions. The LSZ Preview owns that pole mechanism; this page applies it as a calculation workflow and checks why a nonzero free four-point function is not evidence of an interaction.

Required background. LSZ Preview provides pole extraction; Relativistic Normalization fixes external states and amplitudes; From Propagators to Correlators specifies the operator and state data.

Helpful background. Invariant Phase Space provides flux and counting; Generating Functionals distinguishes full and connected correlations.

Specify the scattering observable before the correlator

Use $\hbar=c=1$ and metric $(+---)$. For the ordinary stable-particle construction, choose incoming and outgoing species, momenta, spin states, and an exclusive or inclusive measurement. With covariant external normalization, write

$$\langle f|S-I|i\rangle =i(2\pi)^4\delta^{(4)}(P_f-P_i)\mathcal M_{fi}.$$

The momentum delta is separate from $\mathcal M$. The amplitude also differs from its modulus squared and from a differential cross section. A polarization-resolved calculation keeps specified external spin labels; an unpolarized measurement averages initial labels and sums unobserved final labels at the probability stage.

For each external stable particle choose an interpolating field with a nonzero overlap. The field need not be a fundamental variable of a Lagrangian. Its quantum numbers, pole location, and residue must identify the particle actually being scattered.

The following sequence makes the distinct jobs explicit:

1. Specify the measured channel and the available asymptotic particles.
2. Calculate the relevant vacuum time-ordered field correlation, isolating the connected interaction for an elementary $2\to2$ process.
3. Extract the external one-particle poles and apply the overlap normalization and spin projections specified by LSZ.
4. Fix the overall phase and momentum-delta convention by comparison with $S-I$ above.
5. Form probabilities using the correct flux, spin treatment, phase space, and detector cuts.

For a general multiparticle process there can also be products of independent scattering clusters or spectator propagation. A connected correlator is the building block for a connected interaction, not a rule that every disconnected contribution to every multiparticle $S$ matrix must be discarded.

A free four-point function does not imply scattering

Take a free real scalar field in its vacuum, with zero one-point function, and let $D_{ij}=\langle0|T\phi(x_i)\phi(x_j)|0\rangle$. Gaussian factorization gives

$$C_4(x_1,x_2,x_3,x_4) =D_{12}D_{34}+D_{13}D_{24}+D_{14}D_{23}.$$

This is generally nonzero. However its connected part is

$$C_{4,c}=C_4- \bigl(D_{12}D_{34}+D_{13}D_{24}+D_{14}D_{23}\bigr)=0.$$

For a free $2\to2$ process, the allowed incoming-to-outgoing contractions carry each particle through unchanged. For identical bosons there are two label pairings, reflecting the symmetrized state convention. They belong to the identity part of $S$, with the matching momentum deltas, rather than to a connected interaction amplitude. Pairings incompatible with positive-energy in/out kinematics do not create an extra scattering channel.

The source-language check is equally short. For a centered Gaussian vacuum, $W[J]=-i\log Z[J]$ is quadratic. Its fourth derivative therefore vanishes, whereas the fourth derivative of $Z[J]$ does not. The latter contains the three pair products above. This is why differentiating a source functional four times is not by itself a calculation of a $2\to2$ interaction.

The conclusion depends on the free Gaussian vacuum. An occupied non-Gaussian state can have connected higher moments even under a free Hamiltonian, as the number-state correlator example shows. Such a preparation is not a new vacuum interaction vertex.

Keep the amputation convention attached to the result

The canonical pole calculation distinguishes two operations. If an external scalar overlap is $\sqrt Z$, extracting a unit-residue pole requires division by $\sqrt Z$. If the correlator has already been amputated using its full two-point function, the external factor is instead $\sqrt Z$.

Intermediate object	Remaining scalar external factor per leg
Coefficient after removing the unit-residue pole	$1/\sqrt Z$
Coefficient after full-propagator amputation	$\sqrt Z$

These statements use the definitions on the owner page; they are not interchangeable rules for an unnamed “amplitude.” The physical pole mass, overlap, Fourier arguments, and overall $i$ convention must remain attached to the intermediate object. Lehmann, Symanzik, and Zimmermann (1955) establish the underlying reduction framework; Beisert (2025, sections 10.3–10.4) develops the practical amputation distinction.

For Dirac external legs, retain the selected spinors. The on-shell matrix $p\!\!\!/+m=\sum_su_s\bar u_s$ has rank two, so attempting to invert it as a full four-by-four matrix after going on shell loses the required pole information. Off-shell amputation and external-state projection are separate operations. The fermion-field bridge explains why external spinors appear as field matrix elements.

The amplitude still needs a measurement prescription

For an elementary two-particle initial state, the conventional cross-section structure is

$$d\sigma=\frac{\overline{|\mathcal M|^2}}{\mathcal F}\,d\Phi_f, \qquad \mathcal F=4\sqrt{(p_1\cdot p_2)^2-m_1^2m_2^2}.$$

The bar denotes only the spin sums or averages appropriate to the specified measurement. The final-state measure must count each physical state once, including identical-particle factors or an equivalent nonredundant integration domain. Acceptance cuts and unresolved channels are additional measurement data. The phase-space owner derives these factors and their two-body limit.

A prescribed static background is a different calculation. It supplies no normalized quantum target leg, and it need not conserve the projectile’s spatial momentum. The Mott calculation uses that fixed-source setup. One must explicitly take a heavy-target limit and remove the target-state normalization before comparing it with a dynamical two-body amplitude.

An isolated unstable resonance cannot be used as an exact stable external state in this construction. In charged QED, arbitrarily soft photons also obstruct the naive isolated charged-particle pole and finite free-Fock-state picture. Inclusive observables or suitable dressed states require additional infrared analysis (Buchholz, 1986). The simple LSZ workflow is therefore a conditional reduction, not a proof that all theories have the same asymptotic states.

Exercises

A misleading free calculation. A calculation reports a nonzero four-point function and concludes that the free scalar theory scatters. Identify the missing subtraction and its physical meaning.

Solution

Subtract the three products of two-point functions to obtain $C_{4,c}$. It vanishes in the free vacuum. The surviving kinematically allowed in/out pairings in the full function represent unchanged propagation and identical-state labeling. They reproduce the identity contribution, not an interaction in $S-I$.

A normalization-sensitive number. A program returns a fully amputated four-scalar coefficient with common residue $Z=1/9$. What external overlap factor remains?

Solution

There are four factors $\sqrt Z=1/3$, so the result is multiplied by $1/81$. The overall amplitude phase must still match the program’s convention. Using four factors $1/\sqrt Z$ at this stage would double-count the residue removal.

From an amplitude to a detector count. List three additional specifications needed when $\mathcal M$ is known for a spinful process but the experiment measures an unpolarized angular bin.

Solution

Specify the initial spin average and final spin sum; the invariant flux and final-state measure with correct state counting; and the angular acceptance and other cuts. Converting a cross section into an expected event count additionally requires integrated luminosity and detection efficiency.

References

• Beisert, Niklas. Quantum Field Theory I. ETH Zurich lecture notes, autumn semester 2025, sections 10.3–10.4. Lecture notes. Correlation functions, particle poles, and amputation.
• Buchholz, Detlev. “Gauss’ Law and the Infraparticle Problem.” Physics Letters B 174, 331–334 (1986). doi:10.1016/0370-2693(86)91110-X. Infrared limitations of a sharp charged-particle mass-shell picture.
• Lehmann, Harry, Kurt Symanzik, and Wolfhart Zimmermann. “Zur Formulierung quantisierter Feldtheorien.” Il Nuovo Cimento 1, 205–225 (1955). doi:10.1007/BF02731765.