advanced28 min studyLesson 46 in Physics

Feynman rules as an amplitude language

Translate a specified Lagrangian into algebra, track combinatorics, and distinguish a graph from an observed sequence of events.

Before you begin

Interactions and perturbation theory

On this page

The question for today

What you’ll understand

Obtain the quartic scalar vertex from Wick-contraction counting.
Separate iM from M in a stated S-matrix convention.
Identify external lines, internal momentum integration, and graph symmetry factors.

Feynman diagrams compress a large operator expansion into a visual grammar. Their value is that a diagram corresponds to a definite algebraic term, with defined momenta, couplings, integrals, and counting factors. Their danger is that a familiar-looking sketch can be mistaken for a movie of particles. We will read diagrams as calculations and check them against one explicit interaction.

Begin with a Lagrangian and conventions

Use $\hbar=c=1$ , metric $(+---)$ , and $\mathcal L=\tfrac12(\partial\phi)^2-\tfrac12m^2\phi^2-\lambda\phi^4/4!$ . The Feynman propagator carries $i/(p^2-m^2+i\epsilon)$ . For connected scattering, define the invariant amplitude by stripping $i(2\pi)^4\delta^{(4)}(P_f-P_i)$ from the transition matrix element, using relativistically normalized external states. This fixes whether the symbol being calculated is $i\mathcal M$ or $\mathcal M$ .

At first order, the action expansion contributes $-i\lambda/4!$ times an integral of four fields. Connecting four labeled external scalar legs to that insertion gives $4!$ assignments. Their cancellation with the factorial produces the vertex factor $-i\lambda$ . This is why the factorial belongs in the chosen interaction definition. A different coefficient convention changes the vertex rule, even though the corresponding physical theory can be the same after parameter translation.

\langle f|S-1|i\rangle=i(2\pi)^4\delta^{(4)}(P_f-P_i)\mathcal M_{fi},\qquad i\mathcal M_{2\to2}^{\mathrm{tree}}=-i\lambda,\quad \mathcal M_{2\to2}^{\mathrm{tree}}=-\lambda

The external free states are normalized by ⟨p|q⟩ = 2Eₚ(2π)³δ³(p−q). This expression is for a connected nontrivial transition with the overall conservation delta stripped off.↔ Scroll wide equations horizontally.

Momentum flows are algebraic bookkeeping

Assign a momentum direction to each internal line and impose four-momentum conservation at every vertex. Reversing an arbitrary internal orientation is harmless if its algebra changes consistently. External on-shell momenta obey $p^2=m^2$ ; an internal integration variable generally does not. “Off shell” describes that integration variable and is not a license to violate overall conservation or borrow an experimentally detectable amount of energy for a short time.

A closed loop leaves an independent momentum integration. The loop variable ranges over all allowed four-momenta, so large momenta may make the integral divergent and require a regulator. A diagram symmetry factor counts equivalent contractions not already removed by the expansion coefficients. It is not the same issue as dividing a final-state phase-space integral by a factorial for identical detected particles. Both are counting corrections, but they act at different stages.

\text{internal scalar line: }\frac{i}{k^2-m^2+i\epsilon},\qquad \text{independent loop: }\int\frac{d^4\ell}{(2\pi)^4}

These factors become part of a full graph expression together with vertex factors, conservation conditions, and any graph symmetry factor. A loop integral alone is not an observable.↔ Scroll wide equations horizontally.

Which graphs contribute to which quantity?

Vacuum bubbles have no external legs. In normalized vacuum correlation functions they cancel against the vacuum-to-vacuum normalization. Connected correlation functions exclude products of disconnected pieces; scattering amplitudes additionally require the appropriate external-particle extraction. Amputating external propagator factors is therefore a defined operation, not a decision to erase inconvenient lines. In an interacting theory, the precise relation involves pole residues and the LSZ reduction formula.

Different diagrams for the same final outcome must be added as amplitudes before taking the modulus squared. Which path was taken is not generally available as a separate measured fact. Interference can increase or decrease the resulting probability. Counting the number of drawings or assigning each drawing an independent classical probability loses both relative phases and exchange effects. A good calculation annotates each diagram with algebra before trying to interpret its picture.

Work it through

Two ways to normalize one quartic interaction

Theory A writes $\mathcal L_{\mathrm{int}}=-\lambda\phi^4/4!$ . Theory B writes the same interaction as $-g\phi^4$ . Compare the vertex factors and determine when their leading 2→2 amplitudes agree.

For A, the first-order insertion has coefficient $-i\lambda/4!$ , and $4!$ external assignments leave the vertex factor $-i\lambda$ .
For B, the insertion coefficient is $-ig$ without a compensating factorial. The same assignments therefore leave $-i(24g)$ , giving $\mathcal M_B=-24g$ .
The Lagrangians describe the same interaction coefficient when $g=\lambda/24$ . Under that translation, both vertex factors and both leading scattering amplitudes agree. Equating g with λ would instead compare different interaction strengths.

The result

A Feynman rule is inseparable from the definition of its coupling. An apparent factor-of-24 disagreement can be entirely conventional, but only after the conventions are translated.

The living notebook

Explore the model

Interactive lab

Interactive lab · scattering

A diagram is a term, not a trajectory

Pause the schematic term diagram and vary its toy coupling and angle. Read the explicitly dimensionless g⁴(1 + cos²θ) weight, then compare its angular dependence with the constant leading φ⁴ amplitude derived here.

Incoming termsOutgoing termsAngle θ

Coupling factor g⁴: 0.24
Angular factor 1 + cos²θ: 1.329
Illustrative weight: 0.319

Coupling g0.7Scattering angle θ55°

t = 0 s

A schematic term diagram with dimensionless weight g⁴(1 + cos²θ). Lines denote terms, not particle paths; this is not a process cross-section or loop calculation. Parameter changes pause playback and recompute the model at the current elapsed time. Reset restores both parameters and returns time to zero. Playback starts paused. Adjust either parameter, then choose Play or Step.

Look forThis generic diagram does not implement φ⁴ Feynman rules or evaluate a loop. Its different angular shape is deliberate: a picture needs a stated algebraic model before it can predict a process.

Pause & reason

Check your understanding

Interactive checks need JavaScript. You can still read the explanation.

Read the checkpoint explanation

The diagram gives iM in this convention. Removing its overall i leaves M = −λ. The squared amplitude is λ² at this order, but it belongs in a later observable calculation.

Your turn

Try it yourself

For on-shell elastic scattering of four identical scalars, define $s=(p_1+p_2)^2$ , $t=(p_1-p_3)^2$ , and $u=(p_1-p_4)^2$ . Show that $s+t+u=4m^2$ .

Take a moment to make a prediction or work it out on paper. Open a hint when you need one.

Give me a hint

Expand the three squares and use $p_1+p_2=p_3+p_4$ .

Compare with the solution

The individual mass terms contribute $6m^2$ . The cross terms combine to $2p_1\cdot(p_2-p_3-p_4)=-2p_1^2=-2m^2$ . Their sum is $4m^2$ . This identity checks external momentum assignments; it does not force an internal propagator’s momentum onto the mass shell.

Keep the idea

Take it with you

Derive vertex factors from the written interaction and its combinatorics.
Internal lines and loops encode integration kernels rather than observed paths.
Combine coherent graph amplitudes before forming an observable probability.

Read further

Original sources and deeper explanations to follow at your own pace.

David Tong: Wick’s theorem, diagrams, and scalar amplitudes

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