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. LSZ Preview derives the pole mechanism. Here we use it to organize a scattering calculation and explain why a nonzero free four-point function need not describe 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 linked derivation; 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 calculation 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

Turn on a quartic interaction. Add $\mathcal L_{\rm int}=-\lambda\phi^4/4!$ to the free real scalar theory. At first order, how many Wick contractions connect four labeled external insertions to the four fields at the interaction point? Find the connected contribution and the tree-level invariant amplitude, using unit pole residues.

Solution

There are $4!$ assignments, canceling the denominator in the interaction. With $D_F=\langle T\phi\phi\rangle_0$, the contribution is

$$C_{4,c}^{(1)}(x_1,\ldots,x_4) =-i\lambda\int d^4z\,\prod_{a=1}^4D_F(x_a-z).$$

Removing the four external propagators leaves $i\mathcal M=-i\lambda$, so $\mathcal M=-\lambda$ in the stated $S-I$ convention. Unlike the free pair products, this term joins all four insertions through one interaction.

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.