LSZ and tree amplitudes

A connected time-ordered correlator is not yet a scattering amplitude, and a scattering amplitude is not yet a cross section. LSZ reduction supplies the first bridge by isolating the poles of stable external particles, removing their propagators, and dividing by their field-overlap residues. Flux, phase space, quantum-number sums, and measurement choices supply the second bridge from the resulting S-matrix element to a physical rate.

The running example is elastic scattering in four-dimensional massive real $\lambda\phi^4$ theory. It is simple enough that the pole, normalization, factorial, and dimensional checks can all be completed explicitly, while the same logic extends to spinor and vector external states.

Required background. Canonical quantization and the free scalar fixes relativistic state normalization and the free pole residue. Perturbative expansion and Feynman rules derives the connected quartic correlator and its $-i\lambda$ vertex. Symmetry, currents, and Ward identities supplies the contact-term identity that becomes an on-shell polarization check.

Helpful background. If Fourier residues or delta distributions are uncertain, use the focused Fourier, distributions, and Green-functions review before taking an on-shell limit.

A stable one-particle pole is the entrance to LSZ

Use the $(+---)$ metric, the Minkowski vacuum, and relativistically normalized one-particle states:

$$\langle\mathbf p'|\mathbf p\rangle =(2\pi)^3\,2E_{\mathbf p}\, \delta^{(3)}(\mathbf p'-\mathbf p).$$

The corresponding completeness measure is

$$\int\frac{\mathrm d^3\mathbf p}{(2\pi)^3\,2E_{\mathbf p}}\, |\mathbf p\rangle\langle\mathbf p|.$$

Let a Hermitian interpolating field $\phi$ overlap a stable particle of physical mass $m$:

$$\langle\Omega|\phi(0)|\mathbf p\rangle=\sqrt Z, \qquad Z>0.$$

Inserting a complete set of states into the time-ordered two-point function then produces an isolated one-particle contribution

$$\widetilde G_2(p) =\frac{iZ}{p^2-m^2+i0} +\text{terms less singular at }p^2=m^2.$$

The pole position gives the physical mass; its residue contains the overlap of this particular field with the normalized state. A composite operator is an equally valid interpolating field if its overlap is finite and nonzero. A field rescaling changes $Z$, but the S-matrix cannot depend on that arbitrary choice. Weinberg derives this pole factorization and its arbitrary-spin extension in Weinberg 1995, §§ 10.2–10.3, pp. 430–441.

The words “stable” and “isolated” do real work. Ordinary LSZ assumes a positive-norm one-particle state, a simple real pole separated from the relevant continuum singularities, in/out wave-packet limits, and sufficiently short-ranged asymptotic interactions. A denominator that merely resembles a propagator is not enough.

The joint external residue defines the amplitude

Fix the S-matrix convention

$$S=1+iT,$$

and define the connected invariant amplitude by

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

The identity part of $S$ and any disconnected spectator overlaps are not part of this connected $\mathcal M$. Plane waves are delta-normalized distributions; wave packets or an equivalent finite-volume and long-time limiting argument prevent the momentum delta function from being naively squared. S-matrix and T-matrix normalization develops the convention translation and spectator terms.

For the physical process

$$p_1+p_2\longrightarrow p_3+p_4,$$

take every $p_i^0>0$ on shell. A correlator calculation often uses all momenta incoming, so define

$$q_1=p_1,\qquad q_2=p_2,\qquad q_3=-p_3,\qquad q_4=-p_4.$$

Near a simultaneous set of scalar external poles, the connected Fourier correlator factorizes as

$$\begin{aligned} \widetilde G_c^{(N)}(\{q_r\}) \sim{}&(2\pi)^4 \delta^{(4)}\!\left(\sum_r q_r\right) \\ &\times \left[ \prod_{r=1}^{N} \frac{i\sqrt{Z_r}}{q_r^2-m_r^2+i0} \right] i\mathcal M(\{q_r\}). \end{aligned}$$

This equation makes the reduction operation visible. Strip the single overall delta function and call the remainder $\overline G_c^{(N)}$. Then

$$i\mathcal M =\lim_{q_r^2\to m_r^2} \left[ \prod_{r=1}^{N} \frac{q_r^2-m_r^2}{i\sqrt{Z_r}} \right] \overline G_c^{(N)}.$$

Multiply by every inverse pole before taking any on-shell limit. The $Z_r^{-1/2}$ factors remove the interpolating-field overlaps. The wave-packet derivation and the original reduction formula are developed in Schwartz 2014, § 6.1, pp. 70–74 and Lehmann, Symanzik, and Zimmermann 1955, pp. 205–225.

These related objects should not be collapsed:

Object	What it contains	Ready for a rate formula?
Full correlator	Connected and disconnected time-ordered products, with external poles	No
Connected correlator	Spectator and vacuum-disconnected pieces removed	No
Amputated connected function	External propagator factors removed, often still off shell	No
S-matrix amplitude $\mathcal M$	Joint on-shell residue with external-state normalization fixed	No
Differential cross section or rate	$	\mathcal M

An amputated off-shell Green function is useful, but it is not automatically an S-matrix element. Conversely, LSZ does not amputate a deliberately retained local operator insertion.

The quartic correlator reduces to a constant tree amplitude

Consider

$$\mathcal L =\frac12\partial_\mu\phi\,\partial^\mu\phi -\frac12m^2\phi^2 -\frac{\lambda}{4!}\phi^4, \qquad m>0,\quad \lambda>0.$$

At first order in $\lambda$, the connected four-point correlator contains

$$\widetilde G_{c,\mathrm{tree}}^{(4)} =(2\pi)^4\delta^{(4)}\!\left(\sum_{r=1}^4q_r\right) (-i\lambda) \prod_{r=1}^{4} \frac{i}{q_r^2-m^2+i0}.$$

The interaction factor $1/4!$ has already been canceled by the $4!$ Wick contractions that attach four labeled external fields to the vertex. There is no additional diagram symmetry factor for this labeled contact contribution. At tree level, the canonically normalized scalar has $Z=1$. Applying the four inverse-pole factors leaves

$$i\mathcal M_{\mathrm{tree}}=-i\lambda, \qquad \mathcal M_{\mathrm{tree}}=-\lambda.$$

The overall conservation delta function is not part of the stripped $\mathcal M$. The result passes five immediate checks:

• dimension: in four dimensions, $[\phi]=1$ and $[\lambda]=0$, so the $2\to2$ amplitude is dimensionless;
• identical-particle symmetry: exchanging any external scalar leaves the result unchanged;
• crossing: the same constant analytic function describes the crossed assignments, with the physical regions changed appropriately;
• pole structure: a local contact vertex has no exchange pole; and
• reality at this order: real $\lambda$ gives a real tree-level $\mathcal M$, while the Feynman-rule object remains $i\mathcal M=-i\lambda$.

Crossing is an analytic continuation between physical regions, not permission to relabel momenta without tracking which are incoming, outgoing, and on shell.

Flux and phase space produce the cross section

For total incoming momentum $P=p_1+p_2$ and labeled final momenta, define

$$\mathrm d\Phi_n(P) =(2\pi)^4\delta^{(4)} \left(P-\sum_{j=1}^n p_j\right) \prod_{j=1}^n \frac{\mathrm d^3\mathbf p_j}{(2\pi)^3\,2E_j}.$$

The invariant two-particle flux is

$$\mathcal F =4\sqrt{(p_1\cdot p_2)^2-m_1^2m_2^2}.$$

For a labeled phase-space domain, the master formula is

$$\mathrm d\sigma_{2\to n} =\frac{1}{\mathcal F} \frac{1}{S_f}\, \overline{|\mathcal M_{fi}|^2}\, \mathrm d\Phi_n.$$

$S_f=\prod_a n_a!$ removes permutations of identical final particles that the chosen integration domain counts separately. It is not the factorial in the interaction and not a factor associated with the two prepared incoming beams. The bar denotes exactly the sums over unobserved final labels and averages over unprepared initial labels required by the measurement. Our real scalar example has no spin or color labels.

In the center-of-mass frame,

$$\mathcal F=4\sqrt s\,|\mathbf p_i|,$$

and two-body phase space reduces to

$$\mathrm d\Phi_2 =\frac{1}{16\pi^2} \frac{|\mathbf p_f|}{\sqrt s}\, \mathrm d\Omega.$$

Consequently,

$$\frac{\mathrm d\sigma}{\mathrm d\Omega} =\frac{1}{64\pi^2s} \frac{|\mathbf p_f|}{|\mathbf p_i|} \frac{1}{S_f}\, \overline{|\mathcal M|^2}.$$

For elastic scattering of equal-mass real scalars, $|\mathbf p_f|=|\mathbf p_i|$, $\mathcal M=-\lambda$, and the full labeled solid angle counts the two identical final particles twice. Thus $S_f=2!$ and

$$\boxed{ \frac{\mathrm d\sigma_{\mathrm{tree}}}{\mathrm d\Omega} =\frac{\lambda^2}{128\pi^2s} }.$$

The amplitude is angle independent, so integration over $4\pi$ gives

$$\boxed{ \sigma_{\mathrm{tree}} =\frac{\lambda^2}{32\pi s} }.$$

Equivalently, integrate over one permutation-ordered hemisphere and omit the $1/2!$; using both prescriptions would double-correct the rate. Dimensionally, $[\mathcal M]=0$, $[\mathrm d\Phi_2]=0$, and $[\mathcal F]=2$, so $[\sigma]=-2$, as a cross section must in natural units. The state, flux, and phase-space normalization used here is derived in Schwartz 2014, § 5.1, pp. 57–63 and Srednicki 2007, § 11, pp. 93–101. For general multiplicity and decay kinematics, continue with Lorentz-invariant phase space and Cross sections and decay rates.

This tree-level cross section is a genuine prediction of the declared massive model at leading order, but an amplitude alone is not always a measured observable. Detector cuts, initial-state structure, unstable-particle reconstruction, and infrared-safe inclusive sums can be essential in less idealized theories.

Tree amplitudes have analytic and symmetry checks

A complete tree result should survive checks that do not depend on how the diagrams were drawn:

Check	Expected result
Momentum and mass shell	One overall conservation delta before stripping; every external momentum on its declared shell
Dimension	Couplings, propagators, and momentum powers give the expected dimension of $\mathcal M$
Identical-particle exchange	The full amplitude has the statistics required by the external state
Crossing	Analytic continuation relates crossed processes with all spin, charge, and momentum conventions translated
Physical poles	Exchange poles occur only in allowed channels and at the exchanged mass
Factorization	A tree-pole residue is the product of lower-point on-shell amplitudes, summed over physical intermediate states
Ward replacement	Replacing a massless-vector polarization by its momentum annihilates the complete physical amplitude

For example, an exchanged scalar of mass $M$ produces a denominator $(s-M^2+i0)^{-1}$ in an allowed $s$ channel. Near the pole, the diagram factorizes as

$$i\mathcal M_4 \sim (i\mathcal M_3^{\mathrm L}) \frac{i}{s-M^2+i0} (i\mathcal M_3^{\mathrm R}).$$

The residue check should be performed in the declared $i\mathcal M$ convention, where every factor of $i$ remains visible. A missing crossed graph can violate identical-particle symmetry even if each retained graph has a plausible pole. Scalar contact and exchange amplitudes works through this comparison with all three Mandelstam channels.

Vector external states require Ward and polarization checks

For one external photon of null momentum $k$, write

$$\mathcal M =\varepsilon_\mu(k;q)\,\mathcal A^\mu, \qquad k\cdot\varepsilon=0,$$

where $q$ chooses a polarization representative. Physical representatives obey

$$\varepsilon_\mu\sim\varepsilon_\mu+c\,k_\mu.$$

Reference independence therefore requires

$$k_\mu\mathcal A^\mu=0.$$

This Ward replacement is a check on the complete on-shell amplitude, not on each diagram separately. Contact interactions and crossed graphs can be essential to the cancellation. The time-ordered Ward identity on the previous page contains insertion contacts; under LSZ, those terms combine with inverse external propagators and the on-shell equations to produce the amplitude identity. Dropping them before the joint residue is taken is unsafe.

For an unpolarized rate, sum over the two physical photon helicities. A covariant replacement of that sum by $-\eta_{\mu\nu}$ is legitimate only after the Ward identity has shown that its longitudinal and reference-dependent terms decouple. The physical-state and amplitude checks are developed in Schwartz 2014, §§ 8.4 and 9.4, pp. 123–127 and 147–149 and Vector external states and Ward checks.

Ordinary LSZ has a definite boundary

Use a different observable or asymptotic framework when its hypotheses fail:

• Unstable particles: a resonance pole is generally on an analytically continued sheet at complex invariant mass. It is not an exact asymptotic ket and should occur internally or through a controlled resonance approximation.
• Infraparticles and long-range forces: a charged excitation accompanied by unscreened massless radiation can have branch-point spectral behavior rather than an isolated pole. Dressed-state or inclusive constructions may replace the ordinary Fock-space S-matrix.
• Confined fields: a gauge-fixed quark or gluon propagator does not create a physical colored asymptotic state. Scattering states must belong to the physical spectrum.
• Nonstationary or curved backgrounds: if past and future do not share a common particle notion, in-in correlators, detector responses, or Bogoliubov data may be the appropriate observables.
• Thermal or finite-density states: medium spectral functions and real-time response generally replace vacuum in/out scattering.

The original LSZ theorem does not assert an ordinary S-matrix in any of these settings. LSZ reduction: poles, residues, and stable external states develops the successful hypotheses and the failure branches separately.

Common pitfalls

Deleting external lines instead of taking a residue. Diagrammatic amputation is shorthand. Exact LSZ multiplies by each inverse pole, divides by $\sqrt Z$, and only then takes the joint on-shell limit.

Putting a momentum on shell too early. Setting $p^2=m^2$ before multiplying by $p^2-m^2$ creates an undefined zero-times-pole expression.

Leaving the conservation delta inside $\mathcal M$. The connected amplitude convention strips exactly one overall four-momentum delta distribution.

Confusing three different factorials. The $1/4!$ in the action, any remaining diagram symmetry factor, and the $1/2!$ for overcounted identical final phase space answer different counting questions.

Using an unstable or gauge-dependent field as an external particle. A physical isolated stable pole and a positive-norm asymptotic state are required.

Calling every finite amplitude an observable. A rate also needs flux, phase space, state sums or averages, and a finite measurement definition.

Using a covariant polarization sum before the Ward check. Unphysical components can hide a missing diagram or wrong sign. Test $\varepsilon\to k$ on the complete amplitude first.

Exercises

1. Let $\phi'=c\phi$ with real nonzero $c$. Determine how the pole residue, the connected $N$-point correlator, and the LSZ factors transform. Show that the S-matrix amplitude is unchanged.

   Solution

   The one-particle overlap becomes $\langle\Omega|\phi'|p\rangle=c\sqrt Z$, so $Z'=c^2Z$. An $N$-point correlator containing only $\phi'$ acquires $c^N$. Each external LSZ factor becomes

   $$\frac{p^2-m^2}{i\sqrt{Z'}} =\frac{1}{|c|} \frac{p^2-m^2}{i\sqrt Z}.$$

   For $c>0$, the $N$ inverse factors cancel $c^N$ exactly. For $c<0$, the remaining signs are the consistent phase convention for each one-particle state and do not change probabilities or convention-matched amplitudes. Thus a field normalization changes correlators and $Z$, but not the physical S-matrix.

2. Starting from the quartic connected correlator, recover the tree $\phi\phi\to\phi\phi$ amplitude and total cross section. Identify every factorial and check the dimensions.

   Solution

   The $4!$ attachments of four labeled external fields cancel the $1/4!$ in $-\lambda\phi^4/4!$, leaving the vertex $-i\lambda$. Four tree-level LSZ factors cancel the four external propagators, so $i\mathcal M=-i\lambda$ and $\mathcal M=-\lambda$. No diagram symmetry factor remains for this labeled contact graph.

   For equal-mass elastic scattering, $|\mathbf p_f|/|\mathbf p_i|=1$. The labeled full-sphere phase space counts the two identical final scalars twice, so $S_f=2!$:

   $$\frac{\mathrm d\sigma}{\mathrm d\Omega} =\frac{\lambda^2}{128\pi^2s}.$$

   Integrating over $4\pi$ gives

   $$\sigma=\frac{\lambda^2}{32\pi s}.$$

   The coupling and amplitude are dimensionless in four dimensions, while $1/s$ has dimension $-2$, the correct cross-section dimension. The identical-final $1/2!$ belongs to the rate, not the amplitude or interaction vertex.

3. An amplitude with one external photon is $\mathcal M=\varepsilon_\mu(k;q)\mathcal A^\mu$. Explain why $k_\mu\mathcal A^\mu=0$ is both a reference-independence check and a prerequisite for using a covariant polarization sum in a rate.

   Solution

   Changing the polarization reference at fixed physical helicity shifts the representative by $\varepsilon_\mu\mapsto\varepsilon_\mu+c\,k_\mu$. The amplitude changes by $c\,k_\mu\mathcal A^\mu$, so it is reference independent exactly when the Ward contraction vanishes.

   A physical polarization sum differs from $-\eta_{\mu\nu}$ by terms containing at least one factor of $k_\mu$ or $k_\nu$ and a reference vector. Those terms vanish against an amplitude satisfying the Ward identity. If the contraction fails, replacing the physical sum by $-\eta_{\mu\nu}$ would include unphysical modes and could conceal the failure. The test must be applied to the complete on-shell diagram sum.

4. A two-point function behaves near $p^2=m^2$ as $(p^2-m^2+i0)^{-1+\alpha}$ with nonzero $\alpha$. Why does ordinary scalar LSZ fail?

   Solution

   The singularity is a branch point rather than an isolated simple pole with finite nonzero residue. Multiplication by $p^2-m^2$ does not leave a finite constant that can be interpreted as $\sqrt Z$ times a stable one-particle state. The ordinary inverse-pole reduction therefore has no justified external leg. One must identify the relevant dressed, inclusive, or non-S-matrix observable instead.

Where to go next

You are ready to continue when you can move from a connected correlator to a joint on-shell residue, state every $S$-matrix convention, recover $\mathcal M=-\lambda$, derive the identical-scalar cross section without mixing its factorials, and apply a Ward replacement before a polarization sum.

Continue to Loops and regularization to add quantum corrections while keeping pole prescriptions and dimensions explicit. If the external-state hypotheses fail, follow the appropriate resonance, infrared, confined-state, thermal, or curved-spacetime treatment instead of forcing an ordinary LSZ formula.

References

• Lehmann, Harry, Kurt Symanzik, and Wolfhart Zimmermann. “Zur Formulierung quantisierter Feldtheorien.” Il Nuovo Cimento 1, no. 1 (1955): 205–225. DOI.
• Schwartz, Matthew D. Quantum Field Theory and the Standard Model. Cambridge: Cambridge University Press, 2014. DOI.
• Srednicki, Mark. Quantum Field Theory. Cambridge: Cambridge University Press, 2007. DOI.
• Weinberg, Steven. The Quantum Theory of Fields. Volume I: Foundations. Cambridge: Cambridge University Press, 1995. DOI.