Physical Poles and Tree-Level Factorization

When a tree-level internal momentum $P$ approaches the mass shell of a stable particle, its propagator supplies a simple pole and the graph separates into two lower-point on-shell subgraphs. The residue is their product, summed over a complete set of physical intermediate states. This factorization is a locality and particle-spectrum check; it does not apply unchanged to an unstable pole or a multiparticle threshold.

Required background. Connected Tree Diagrams and Amputated Amplitudes supplies the internal-propagator structure. Relativistic Scattering Kinematics supplies channel invariants and on-shell limits.

Helpful background. Poles, Cuts, Thresholds, and Stable Particles distinguishes isolated stable poles from continuum singularities.

The scalar pole separates the graph

Suppose a tree graph contains one scalar propagator carrying $P=\sum_{a\in L}p_a$. Everything on its two sides is regular as $P^2\to m^2$. With the chapter convention,

$$i\mathcal M_n \xrightarrow[P^2\to m^2]{} (i\mathcal M_L) \frac{i}{P^2-m^2+i0} (i\mathcal M_R)+O(1).$$

For the cubic scalar example, $i\mathcal M_L=i\mathcal M_R=-ig$, so

$$\mathcal M_4 =-\frac{g^2}{P^2-m^2+i0}+O(1).$$

The factorization is not an approximation to the entire amplitude: it is the leading singular term in a specified channel. Contact graphs and exchange graphs in other channels contribute to the regular $O(1)$ part.

The pole may lie outside the real physical region of the channel being measured. Factorization is an analytic statement about the residue; physical production additionally requires compatible quantum numbers and kinematics. Schwartz derives the single-particle pole and lower-point factorization in Schwartz 2014, § 24.3, printed pp. 471–474.

Sum over the intermediate physical states

If the exchanged particle has spin or other discrete labels, insert its one-particle completeness relation:

$$i\mathcal M_n \longrightarrow \sum_{\lambda,a} (i\mathcal M_L^{\lambda,a}) \frac{i}{P^2-m^2+i0} (i\mathcal M_R^{\lambda,\bar a}).$$

This compact notation pairs the state inserted on the left with its dual on the right. Equivalently, the internal momentum is $P$ in one subamplitude and $-P$ in an all-incoming convention, and a charged representation $a$ is paired with its conjugate $\bar a$. Those crossings can carry convention dependent phases or fermionic signs; they are part of the definitions of $\mathcal M_L$ and $\mathcal M_R$, not an extra factor guessed after the state sum.

For a fermion, the numerator identity

$$P\!\!\!/+m=\sum_s u_s(P)\bar u_s(P)$$

at $P^0>0$ converts the propagator numerator into the particle spin-state sum and splits the spinor chain. If the chosen orientation instead carries a positive-energy antifermion, use $\sum_s v_s(P)\bar v_s(P)=P\!\!\!/-m$ with a consistently reversed momentum assignment; mixing the two orientations is a common sign error. For a massive vector,

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

For a massless gauge boson, a covariant numerator contains unphysical pieces. Only after the lower-point amplitudes obey their Ward identities can those pieces be replaced by a sum over physical helicities. Thus factorization and gauge consistency reinforce one another: a residue that depends on a gauge reference signals an incomplete or inconsistent subamplitude.

If $A_L^\mu$ and $A_R^\nu$ are the two on-shell lower-point currents, the physical helicity sum may be written

$$\sum_{h=\pm} (A_L\cdot\varepsilon_h(P;q)) (\varepsilon_h^*(P;q)\cdot A_R).$$

Changing the reference $q$ adds terms proportional to $P^\mu$ or $P^\nu$. They vanish only if $P_\mu A_L^\mu=P_\nu A_R^\nu=0$. Thus a reference-free residue is an independent check that both lower-point subamplitudes are complete.

Why locality predicts simple tree poles

A local interaction vertex is polynomial in momenta. At tree level, non-polynomial singularities therefore come from internal propagators. A single internal line gives a simple pole, and cutting it disconnects the tree because every edge of a tree is a bridge. These graph and analytic facts are the same statement in two languages. The relation between pole factorization and locality is developed in Schwartz 2014, §§ 24.3–24.4, printed pp. 471–476.

Repeated poles do not arise from one ordinary stable propagator. Apparent higher-order or spurious poles may result from a representation of the amplitude and must cancel or be explained by genuine repeated propagation. Checking every channel residue against lower-point amplitudes is therefore a powerful diagnostic of signs, couplings, state sums, and missing contact terms.

Boundaries of the statement

• A multiparticle intermediate state produces a branch point and cut after loop integration, not a single tree pole.
• An unstable particle is not an asymptotic state with a real physical-sheet pole. Its resonance pole lies on an unphysical sheet and its residue may be complex.
• Self-energy resummation and a hand-inserted width mix loop orders; they are not part of tree factorization.
• Generalized unitarity and BCFW recursion use related factorization data but add complex kinematics or loop-level cut constructions.

These distinctions are developed in Resonance Poles, Riemann Sheets, and Unstable States and the later on-shell and cut chapters. Complex Momenta and Factorization uses these residues at complex momenta, BCFW Recursion adds a large-shift boundary condition, and Generalized Unitarity and Integrand Reduction replaces the single tree pole by loop-level on-shell cut information. The unitarity and pole interpretation behind these distinctions is reviewed in Weinberg 1995, §§ 3.6–3.8, printed pp. 147–165.

Check your understanding

Take a fermion exchange graph, multiply its numerator by the on-shell completeness relation, and mark the two resulting lower-point spinor chains. Then repeat with a massless vector and state the Ward identity needed to discard reference-dependent terms. The check passes only if the intermediate sum uses physical states and the regular remainder is not misidentified as part of the residue.

Solution

At $P^2=m^2$ and $P^0>0$, insert $P\!\!\!/+m=\sum_su_s(P)\bar u_s(P)$ between the matrices on the two sides. Each term is the product of a left chain ending in $u_s(P)$ and a right chain beginning with $\bar u_s(P)$, summed over $s$. For a massless vector, insert the two-helicity projector. Its reference-dependent pieces are proportional to $P_\mu A_L^\mu$ or $P_\nu A_R^\nu$, so they vanish when each on-shell subamplitude satisfies its Ward identity. Contact terms and other channels are regular at the selected pole and belong to $O(1)$, not to the residue.

References

• Schwartz, Matthew D. Quantum Field Theory and the Standard Model. Cambridge University Press, 2014, §§ 24.3–24.4, printed pp. 471–476. doi:10.1017/9781139540940.
• Weinberg, Steven. The Quantum Theory of Fields, Volume I: Foundations. Cambridge University Press, 1995, §§ 3.6–3.8, printed pp. 147–165. doi:10.1017/CBO9781139644167.