LSZ for Spinor and Vector External States

Spin does not change the logic of LSZ: isolate a stable one-particle pole, divide by the square root of its residue, amputate the external propagator, and take the positive-energy on-shell limit. What changes is the pole numerator. A Dirac leg ends in an on-shell $u$ or $v$ spinor; a spin-one leg ends in a physical polarization vector. Gauge-fixed vector components that do not represent positive-norm asymptotic states are never promoted to external particles.

The common pole-factorization logic for arbitrary spin is developed in Weinberg 1995, §§ 10.2–10.3, pp. 430–441.

Required background. LSZ Reduction: Poles, Residues, and Stable External States supplies the scalar residue argument. The Fermion Propagator fixes the Dirac pole numerator. Massive and Massless Spin-One Polarizations fixes the physical polarization spaces.

Helpful background. Covariant Free-Photon Quantization and Propagator distinguishes gauge-fixed propagator components from physical photon states.

Dirac poles and external spinors

Use

$$p\!\!\!/\equiv\gamma^\mu p_\mu, \qquad \{\gamma^\mu,\gamma^\nu\}=2\eta^{\mu\nu}.$$

For a stable Dirac particle of mass $m$, choose phases and normalization so that

$$\begin{aligned} \langle\Omega|\psi(0)|p,s\rangle&=\sqrt{Z_2}\,u_s(p),\\ \langle\Omega|\bar\psi(0)|\bar p,s\rangle&=\sqrt{Z_2}\,\bar v_s(p). \end{aligned}$$

Near the physical pole, the exact propagator has the form

$$\widetilde S_F(p) =\frac{iZ_2(p\!\!\!/+m)}{p^2-m^2+i0} +\text{less singular terms}.$$

The numerator is the positive-energy spin projector because

$$\sum_s u_s(p)\bar u_s(p)=p\!\!\!/+m, \qquad \sum_s v_s(p)\bar v_s(p)=p\!\!\!/-m.$$

We use $\bar u_su_{s'}=2m\delta_{ss'}$ and $u_s^\dagger u_{s'}=2E_{\mathbf p}\delta_{ss'}$, with analogous antiparticle relations. The on-shell equations

$$(p\!\!\!/-m)u_s(p)=0, \qquad (p\!\!\!/+m)v_s(p)=0$$

ensure that the inverse Dirac operator amputates the pole and leaves the corresponding wave function. Srednicki derives the fermionic asymptotic overlaps and reduction factors in Srednicki 2007, § 41, pp. 263–267.

For a connected amputated object $\mathcal A$ with its open Dirac indices displayed, the four familiar endpoints are

External particle	State direction	Endpoint contracted with $Z_2^{-1/2}\mathcal A$
fermion	incoming	$u_s(p)$
fermion	outgoing	$\bar u_s(p)$
antifermion	incoming	$\bar v_s(p)$
antifermion	outgoing	$v_s(p)$

The placement is not mnemonic decoration: it follows from whether $\psi$ or $\bar\psi$ has the nonzero vacuum-to-particle matrix element and from the ordering of the open spinor index. Reversing a fermion line or crossing a particle requires the statistics and phase convention of the amplitude, not only replacing $u$ by $v$.

Massive spin-one poles

For a stable massive vector state,

$$\langle\Omega|A_\mu(0)|p,\lambda\rangle =\sqrt{Z_V}\,\varepsilon_\mu(p,\lambda),$$

where

$$p\cdot\varepsilon(p,\lambda)=0, \qquad \varepsilon^*(p,\lambda)\cdot\varepsilon(p,\lambda') =-\delta_{\lambda\lambda'}.$$

There are three physical polarizations, and their completeness relation is

$$\sum_{\lambda=1}^{3} \varepsilon_\mu(p,\lambda) \varepsilon_\nu^*(p,\lambda) =-\eta_{\mu\nu}+\frac{p_\mu p_\nu}{m^2}.$$

After amputating the vector pole and multiplying by $Z_V^{-1/2}$, an incoming leg is contracted with $\varepsilon_\mu(p,\lambda)$ and an outgoing leg with $\varepsilon_\mu^*(p,\lambda)$. The completeness tensor provides an independent check: it is transverse and has trace $-3$ with both indices lowered in the $(+---)$ convention.

An unstable massive vector does not satisfy the premise. A complex resonance pole or narrow line shape can be treated inside a larger stable-particle amplitude, but there is no exact asymptotic $|p,\lambda\rangle$ to reduce.

Massless vectors and the physical-state restriction

For a stable massless gauge boson, only the two transverse helicities are physical external states. Choose a reference vector $n$ with $p\cdot n\neq0$. A useful polarization sum is

$$\sum_{\lambda=1}^{2} \varepsilon_\mu(p,\lambda;n) \varepsilon_\nu^*(p,\lambda;n) =-\eta_{\mu\nu} +\frac{p_\mu n_\nu+n_\mu p_\nu}{p\cdot n} -\frac{n^2p_\mu p_\nu}{(p\cdot n)^2}.$$

Changing $n$ changes the representatives by terms proportional to $p_\mu$. A complete physical amplitude $\mathcal A^\mu$ must therefore satisfy the on-shell Ward check

$$p_\mu\mathcal A^\mu=0,$$

so that $\varepsilon_\mu\to\varepsilon_\mu+\alpha p_\mu$ leaves $\varepsilon_\mu\mathcal A^\mu$ unchanged. The gauge-fixed propagator may contain longitudinal or scalar components, but LSZ contracts its physical pole residue with transverse external polarizations. Srednicki gives the photon reduction and normalization condition in Srednicki 2007, § 56, pp. 339–342.

This page uses the Ward identity as a necessary amplitude check. The general BRST proof that gauge-parameter dependence cancels between physical states belongs to the gauge-structure treatment, and Vector External States and Ward Checks performs the perturbative complete-amplitude test.

One reduction pipeline, different pole numerators

The scalar LSZ map remains valid after replacing its scalar endpoint by a spin projector. In the figure, inspect the final stage: the inverse pole removes propagation, while the spinor or polarization selects a state inside the pole residue.

For a Dirac leg, $p\!\!\!/+m$ or $p\!\!\!/-m$ in the pole residue is resolved by $u$ or $v$ wave functions. For a stable vector leg, the residue is resolved by physical polarizations; gauge-fixed auxiliary directions are excluded. The same isolated-pole, packet, and stability assumptions as scalar LSZ remain in force. Schematic and not to scale.

Field pole	Physical residue data	External contraction	Failure to exclude
scalar	$Z$	$1$	non-simple or nonphysical pole
Dirac	$Z_2(p\!\!\!/+m)$ or antiparticle projector	$u,\bar u,v,\bar v$	unstable or infraparticle charged state
massive vector	$Z_V[-\eta_{\mu\nu}+p_\mu p_\nu/m^2]$	three $\varepsilon_\mu$	unstable vector resonance
massless gauge vector	transverse physical residue	two helicity polarizations	longitudinal gauge modes or confined gauge quanta

A residue and spin-sum check

For a real coupling $g$, suppose an amputated scalar source creates a fermion pair through

$$\mathcal M= g\,\bar u_s(p)v_r(k).$$

Summing over final spins gives

$$\begin{aligned} \sum_{s,r}|\mathcal M|^2 &=g^2\operatorname{tr}\!\left[ (p\!\!\!/+m)(k\!\!\!/-m)\right]\\ &=4g^2(p\cdot k-m^2). \end{aligned}$$

The first line checks the external-state normalization: each spin sum reproduces the numerator of the corresponding on-shell pole. A rate for unobserved final spins sums this expression; it does not divide by the number of final polarizations. Initial-state averaging is a separate specification of the prepared beam ensemble.

Boundaries that survive the formal similarity

• A photon can be a sharp stable external particle while a charged excitation in the same theory fails ordinary Fock-state LSZ because of soft radiation.
• A perturbative gluon polarization is useful inside a partonic calculation, but confinement prevents an exact colored asymptotic S-matrix state.
• A massive Proca field has three physical polarizations; a massless gauge field has two after quotienting gauge directions. Setting $m=0$ in the massive projector is singular and is not a derivation of the massless state space.
• Mixing fields require a matrix residue. One must identify and normalize the physical pole eigenvectors before attaching external wave functions.

For the last point, suppose several Dirac interpolators $\psi_A$ overlap stable states $|p,s,a\rangle$ of the same mass through

$$\langle\Omega|\psi_A(0)|p,s,a\rangle =\zeta_{Aa}u_s(p).$$

State insertion factorizes the pole residue:

$$S_{AB}(p)\sim \frac{i}{p^2-m^2+i0} \sum_a\zeta_{Aa}(p\!\!\!/+m)\zeta^\dagger_{aB}.$$

Choose a dual on the residue image, $\widehat\zeta^{aA}\zeta_{Ab}=\delta^a{}_b$, and contract with $\widehat\zeta^{aA}$ to select the normalized physical state before applying the Dirac inverse. Left-null field combinations have zero overlap and cannot create that particle. This factorized prescription is invariant under changes of interpolating-field basis; taking square roots of individual matrix entries is not.

Common pitfalls

“The numerator of a fermion propagator is amputated and discarded.” Amputation removes the inverse propagator; its on-shell residue is resolved into the external spinor. Dropping that wave function loses spin information.

“Use $-\eta_{\mu\nu}$ for every external vector sum.” For a massive vector the longitudinal physical polarization contributes $p_\mu p_\nu/m^2$. For a massless vector, reference-dependent terms disappear only after contraction with a Ward-consistent amplitude.

“Any pole in a gauge-fixed propagator is an external particle.” Physical-state conditions and positive norm are indispensable. Auxiliary, ghost, and confined excitations do not become external states through LSZ notation.

Check your understanding

1. Verify that $\sum_s u_s\bar u_s=p\!\!\!/+m$ is annihilated by $p\!\!\!/-m$ on shell.

   Answer

   Multiply: $(p\!\!\!/-m)(p\!\!\!/+m)=p^2-m^2=0$. Thus the spin sum lies in the kernel of the on-shell inverse Dirac operator, as a physical pole residue must.

2. Why may a reference vector appear in a massless polarization sum but not in a physical rate?

   Answer

   The reference selects representatives of the two-dimensional gauge quotient. Changing it adds momentum-proportional terms. Contracting with a complete amplitude satisfying $p_\mu\mathcal A^\mu=0$ removes those terms, so the rate is reference independent.

Continue

Vector External States and Ward Checks tests the physical-polarization rule in complete amplitudes. Cross Sections and Decay Rates explains spin sums and initial averages. On-Shell States and Little-Group Scaling repackages the same physical state data without gauge-redundant fields.

References

• 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.