Tree Amplitudes and Gauge Consistency

Tree amplitudes are the first place where perturbative rules become a coherent scattering calculation. The chapter begins with connected, amputated graphs; adds scalar, fermion, and vector external states; and then uses channels, crossing, poles, factorization, and Ward replacements as independent checks. Enter through the scalar benchmark if you want a first calculation, through the fermion or vector pages if signs or gauge redundancy are the obstacle, or through channels and poles if your goal is analytic structure.

The scope is deliberately tree level. Loop topology and branch cuts begin in Loop Calculations and Integral Reduction, cross sections are normalized in Cross Sections and Decay Rates, and a general analytic crossing theorem is not inferred from diagram relabeling.

Check your entry point

You are ready for the construction route if you can do all three of the following:

• turn an interaction monomial into a momentum-space vertex, including its factorial and sign;
• state why LSZ removes external propagators but not internal ones; and
• distinguish the overall momentum delta function from the stripped invariant amplitude.

Repair the first item in Momentum-Space Feynman Rules and the other two in LSZ Reduction: Poles, Residues, and Stable External States. For the vector route, also verify that you can identify the two physical massless spin-one polarizations in Massive and Massless Spin-One Polarizations. For the fermion route, you should be able to distinguish $u$, $\bar u$, $v$, and $\bar v$ external wave functions and use their completeness relations; repair that skill in Plane Waves, Spin Sums, and Bilinears. For the analytic route, check that you can derive $s+t+u=\sum_i m_i^2$ from momentum conservation and on-shell conditions; Relativistic Scattering Kinematics provides the repair.

Choose a route

Goal	Suggested route	Observable result
First complete tree calculation	Connected trees → scalar contact and exchange	Enumerate all graphs, strip external legs, and check dimensions and symmetry.
Fermion signs and decay	Connected trees → fermion and Yukawa amplitudes	Build ordered spinor chains and recover the scalar-to-fermion threshold factor.
Gauge consistency	Connected trees → vector Ward checks	Replace one polarization by its momentum and see the complete amplitude vanish.
Crossing and poles	Scalar exchange → Mandelstam channels → factorization	Separate physical regions, locate simple poles, and factorize their residues.

These are reading routes, not additional prerequisite claims.

The common calculation spine

All six pages use the same sequence:

1. Fix the external states and the $S$, $T$, and $\mathcal M$ normalization.
2. Draw every connected tree topology compatible with the fields and order.
3. Assign momenta consistently and impose one conservation equation per vertex.
4. Multiply vertices, internal propagators, and external wave functions; LSZ has already removed external propagators.
5. Sum the diagrams before testing permutation symmetry, Ward identities, poles, residues, or crossing.
6. Hand $|\mathcal M|^2$ to the rate formula only after spin sums, averages, and identical-particle factors are declared.

A local contact term is polynomial in external momenta. Propagation produces a denominator and therefore a possible pole. Gauge redundancy instead constrains the sum of diagrams: reference-dependent pieces need not vanish separately. These three facts—locality, propagation, and cancellation—are the organizing distinction of the chapter.

The connected-amplitude construction and its gauge extension are developed in Schwartz 2014, §§ 7.3–7.4, 8.4, and 9.2–9.4, printed pp. 93–99, 123–127, and 142–149. Complementary treatments of scalar and fermion graph combinatorics appear in Srednicki 2007, §§ 10–11 and 45–48, printed pp. 87–103 and 282–302, while the scattering and gauge-invariance framework is given in Weinberg 1995, §§ 6.1–6.3 and 8.6–8.7, printed pp. 259–285 and 355–368.

Chapter guide

1. Connected Tree Diagrams and Amputated Amplitudes gives the graph enumeration and normalization algorithm. It is the required entry for all worked tree calculations.
2. Scalar Contact and Exchange Amplitudes contrasts a quartic contact interaction with cubic $s$-, $t$-, and $u$-channel exchange, including identical-particle and limiting checks.
3. Fermion and Yukawa Tree Amplitudes fixes external spinors, fermion-line order, particle/antiparticle flow, spin sums, and a scalar decay example.
4. Vector External States and Ward Checks explains physical polarizations and demonstrates a complete scalar-QED Ward cancellation.
5. Mandelstam Channels and Tree-Level Crossing translates between all-incoming momenta and distinct physical channels without confusing relabeling with an analytic theorem.
6. Physical Poles and Tree-Level Factorization derives the simple-pole residue as a product of lower-point on-shell amplitudes and states the stable-particle and loop limitations.

Conventions that must stay visible

The site-wide metric and Fourier conventions apply. In four dimensions this chapter uses

$$S=1+iT, \qquad \langle f|iT|i\rangle =i(2\pi)^4\delta^{(4)}(P_f-P_i)\mathcal M_{fi},$$

with one-particle normalization

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

Thus $i\mathcal M$, not $\mathcal M$, is the direct product of diagram factors. Crossing pages additionally declare which momenta are treated as incoming. Fermion pages fix external-operator order. Vector pages test the invariant statement $\mathcal M(\varepsilon\to k)=0$, which is unchanged by polarization-basis conventions.

Review the chapter

Use the questions below to identify topics worth revisiting.

• Construction and normalization. Starting from $-g\phi^2\chi/2$, enumerate the four-scalar tree graphs and identify the single overall momentum delta function. A successful response leaves no external propagator or free momentum integral and states whether it is computing $i\mathcal M$ or $\mathcal M$.
• Analytic interpretation. Explain why a $-\lambda\phi^4/4!$ contact graph has no exchange pole, then factorize the residue of a cubic exchange pole. The residue must be expressed as two on-shell three-point factors with the intermediate-state convention visible.
• Representation change. Map one all-incoming four-point expression from the physical $s$-channel to the physical $t$-channel. A complete answer changes the particle or antiparticle assignment and the real physical region, not merely the letters $s$ and $t$.
• Failure diagnosis. Remove the scalar-QED seagull graph and perform one Ward replacement. The nonzero remainder should identify the missing contact term; after it is restored, the complete sum must vanish before any polarization sum is used.
• Enumeration transfer. Count the labeled six-point trees in $\phi^4$ theory. A complete answer uses two quartic vertices joined by one internal line and obtains $\tfrac12\binom63=10$ unordered $3|3$ partitions, each with a different channel invariant and no loop integration.

If a response fails at graph enumeration, return to connected trees. If it fails at channel interpretation, use Mandelstam channels; if it fails at the residue or Ward test, use factorization or vector Ward checks, respectively.

Where to go next

• To turn a checked tree amplitude into a measured rate, continue to Cross Sections and Decay Rates.
• To test the result against probability conservation, continue to S-Matrix Unitarity and The Optical Theorem and Cut Interpretation.
• To use pole residues as constructive input at complex momenta, continue to Complex Momenta and Factorization and then BCFW Recursion.

References

• Schwartz, Matthew D. Quantum Field Theory and the Standard Model. Cambridge University Press, 2014, §§ 7.3–7.4, 8.4, and 9.2–9.4, printed pp. 93–99, 123–127, and 142–149. doi:10.1017/9781139540940.
• Srednicki, Mark. Quantum Field Theory. Cambridge University Press, 2007, §§ 10–11 and 45–48, printed pp. 87–103 and 282–302. doi:10.1017/CBO9780511813917.
• Weinberg, Steven. The Quantum Theory of Fields, Volume I: Foundations. Cambridge University Press, 1995, §§ 6.1–6.3 and 8.6–8.7, printed pp. 259–285 and 355–368. doi:10.1017/CBO9781139644167.