Perturbative expansion and Feynman rules

Feynman rules are compressed instructions for a particular perturbative expansion. Their propagators come from the quadratic action and its boundary conditions; their vertices come from the interaction; their numerical factors come from the exponential and Wick contractions. Deriving one complete set of rules makes signs and symmetry factors checkable instead of mnemonic.

This lesson uses a real scalar with a quartic interaction as the running example, then explains what changes for fermions and gauge fields. It develops time-ordered correlation functions. External-state normalization and rates enter on the next page.

Required background. Functional integrals and correlators supplies the free generating functional and Wick factorization; fermions, spin, and anticommutation supplies graded ordering; and symmetry, currents, and Ward identities supplies the checks an interacting calculation must preserve. For a gauge theory, first complete vector fields and gauge redundancy.

Split the action without changing the theory

Consider

$$S[\phi]=S_0[\phi]+S_{\mathrm{int}}[\phi],$$

with

$$S_0=\int\mathrm d^d x\, \left(\frac12\partial_\mu\phi\,\partial^\mu\phi -\frac12m^2\phi^2\right), \qquad S_{\mathrm{int}}=-\int\mathrm d^d x\, \frac{\lambda}{4!}\phi^4.$$

The split defines the perturbative organization: $S_0$ fixes the free vacuum, propagator, and Wick contractions, while $S_{\mathrm{int}}$ is expanded in $\lambda$. Moving a quadratic term between the two pieces changes the free propagator and the interaction counterterm; it cannot be done in only one place without changing the expansion.

With the normalized vacuum functional, the interacting correlator is

$$\langle\mathrm T\,\mathcal O\rangle = \frac{ \left\langle0\left|\mathrm T\left[ \mathcal O\exp\left(iS_{\mathrm{int}}\right) \right]\right|0\right\rangle_0 }{ \left\langle0\left|\mathrm T\exp\left(iS_{\mathrm{int}}\right) \right|0\right\rangle_0 }.$$

The denominator cancels vacuum bubbles—components with no connection to any insertion in $\mathcal O$. It does not cancel self-energy or tadpole subdiagrams that remain connected to external insertions.

Equivalently, with

$$Z_0[J]=\exp\left[-\frac12 J\mathbin{\cdot}D_F\mathbin{\cdot}J\right],$$

the interacting source functional is

$$Z[J]=\frac{1}{\mathcal N} \exp\left[ -\frac{i\lambda}{4!} \int\mathrm d^d z\, \left(\frac{1}{i}\frac{\delta}{\delta J(z)}\right)^4 \right]Z_0[J], \qquad Z[0]=1.$$

Each source derivative inserts a field, and differentiation of the Gaussian performs all Wick pairings. This is the algebra behind the diagrams.

The quartic vertex comes from counting contractions

Expand the interaction exponential once in the connected four-point function:

$$\frac{-i\lambda}{4!} \int\mathrm d^d z\, \left\langle0\left|\mathrm T\left[ \phi(x_1)\phi(x_2)\phi(x_3)\phi(x_4)\phi(z)^4 \right]\right|0\right\rangle_0.$$

For the contact contribution, each external field contracts with one of the four fields at $z$. There are $4!$ such bijections, which cancel the $1/4!$ in the action. The connected result is

$$G^{(4)}_{c,\mathrm{tree}}(x_1,x_2,x_3,x_4) =-i\lambda\int\mathrm d^d z\, \prod_{a=1}^4D_F(x_a-z).$$

Fourier transformation assigns one momentum to every line. The $z$ integral produces momentum conservation,

$$\int\mathrm d^d z\,e^{-iz\cdot(p_1+p_2+p_3+p_4)} =(2\pi)^d\delta^{(d)}\!\left(\sum_{a=1}^4p_a\right),$$

when all momenta are taken into the vertex. After stripping that delta function, the quartic vertex factor is

$$-i\lambda.$$

The $4!$ in the Lagrangian was chosen so this rule is simple. If the interaction were written $-g\phi^4/4$, the vertex would be $-i(4!/4)g=-i6g$. A remembered vertex factor is therefore meaningless until the action normalization is stated. The expansion and its combinatorics are developed in Schwartz 2014, §§ 7.1–7.4.

Build a momentum-space integrand systematically

For the scalar theory above, a connected diagram contributes the product of the following ingredients:

1. For each internal scalar line of momentum $p$,

   $$\frac{i}{p^2-m^2+i0}.$$

2. For each quartic vertex, $-i\lambda$ and a momentum-conserving delta function with every incident momentum oriented inward.

3. For each independent loop momentum, an integral $\int\mathrm d^d\ell/(2\pi)^d$.

4. A factor $1/S$, where $S$ counts permutations of indistinguishable internal elements that leave the labeled diagram unchanged.

5. Any external propagators required by the correlation function being computed. These are removed and replaced by state normalizations only in the later LSZ step.

Use the vertex delta functions to eliminate redundant momentum integrals, but retain one overall delta function for total momentum conservation. The number of independent loop momenta in a connected graph is

$$L=I-V+1,$$

where $I$ is the number of internal lines and $V$ the number of vertices. This topological relation checks the integral count; it says nothing yet about convergence or regularization.

A symmetry factor is not a correction guessed from the drawing. In the Dyson series, $1/V!$ accounts for permuting identical vertices and factorials in the interaction account for permuting fields at a vertex. Wick contractions cancel most of these factors. Whatever permutations still produce the same labeled contraction pattern form the remaining $S$. Re-derive it whenever fields, external labels, or interaction normalization change. See Srednicki 2007, §§ 9–10 for a systematic contraction-based treatment.

Connected, amputated, and one-particle-irreducible are different

These frequently compressed words answer different questions.

Object	What has been removed or restricted
Full correlator	Nothing; includes disconnected products
Connected correlator	Disconnected products removed by derivatives of $W=-i\log Z$
Amputated correlator	External propagator factors removed
One-particle-irreducible vertex	Cannot be disconnected by cutting one internal line

An amputated connected function is not automatically an observable amplitude. LSZ additionally requires stable asymptotic states, pole residues, on-shell limits, and normalization factors. Likewise, a one-particle-irreducible two-point insertion is not the full propagator; the latter is obtained by resumming such insertions under stated conditions.

Fermions attach signs to ordering

For a Dirac field, an oriented internal line carries

$$\frac{i(p!!!/+m)}{p^2-m^2+i0}.$$

The arrow records fermion-number flow for a Dirac field, not necessarily the direction of momentum. Vertex matrices and coupling signs come from the ordered interaction density. External spinors, their adjoints, and their order must match the declared incoming and outgoing states.

Two sign rules have a common origin: odd objects anticommute.

• Reordering external fermionic operators into a standard order can produce a permutation sign.
• Every closed fermion loop contributes an additional minus sign because the contraction chain must be cyclically reordered to close.

Do not add a second minus sign by visual habit after it has already been included through an ordered functional derivative calculation. A robust check is to derive a low-order correlation function from Grassmann sources, then verify that the diagrammatic rule reproduces it.

Gauge theories need more than vector propagators

A perturbative gauge theory begins only after the redundancy has been handled. The calculation must state the gauge-fixing term, gauge parameter, physical external-state prescription, and—outside a free Abelian theory—the ghost action or equivalent BRST framework. Gauge-dependent propagators and vertices are allowed intermediate objects; physical quantities must satisfy the appropriate Ward or Slavnov–Taylor identities.

At tree level with an external photon, replacing a polarization by its momentum should give zero after all relevant diagrams are summed:

$$\varepsilon_\mu(k)\mathcal M^\mu \longrightarrow k_\mu\mathcal M^\mu=0.$$

A single diagram need not pass that test. Failure after the required sum can signal a missing diagram, a momentum-routing error, an inconsistent vertex, or symmetry breaking by the regulator or approximation. Tuning one sign to force agreement is not a derivation.

Common pitfalls

Importing a rule without its action. Coupling factorials, signs, group generator conventions, and Fourier phases all affect the rule. Start from the declared Lagrangian.

Canceling every vacuum-looking subgraph. Only components disconnected from all external insertions cancel against the normalization denominator. A tadpole attached to an external line is still part of the connected correlator.

Hiding momentum orientation. Choose all momenta inward at each vertex and write the conservation equation. A different convention is fine if every propagator and external state is translated with it.

Guessing a symmetry factor from visual similarity. Count contractions or the automorphisms of the labeled graph. External labels and field species can destroy apparent symmetries.

Calling a Green function an amplitude. Correlators include external propagators and can be gauge dependent. Amputation, pole residues, and asymptotic-state assumptions enter next.

Exercises

1. Recover the quartic contact factor

At first order in $\lambda$, count the Wick contractions that connect four distinct external scalar fields to one $\phi^4$ vertex. Explain the fate of the $1/4!$ and give the momentum-space vertex.

Solution

Choose which of the four vertex fields contracts with $x_1$ in four ways, then with $x_2$ in three ways, $x_3$ in two ways, and $x_4$ in one way. The number is $4\cdot3\cdot2\cdot1=4!$. It cancels the interaction normalization, leaving

$$-i\lambda\int\mathrm d^d z\, D_F(x_1-z)D_F(x_2-z)D_F(x_3-z)D_F(x_4-z).$$

Fourier transforming the four propagators and integrating over $z$ produces the overall delta function. The stripped vertex factor is $-i\lambda$ with all four momenta oriented inward.

2. Derive the one-vertex tadpole factor

At first order in $\lambda$, find the connected contraction in the two-point function where $x$ and $y$ attach to one quartic vertex and the remaining two vertex fields contract with each other. Determine its numerical factor.

Solution

There are four choices for the vertex field paired with $\phi(x)$ and three remaining choices for the one paired with $\phi(y)$. The final two fields then pair uniquely, giving $12$ contractions. Multiplication by $(-i\lambda)/4!$ leaves

$$-\frac{i\lambda}{2} \int\mathrm d^d z\, D_F(x-z)D_F(z-z)D_F(z-y).$$

The factor $1/2$ is the diagram’s symmetry factor. The coincident propagator $D_F(z-z)$ is ultraviolet singular in the continuum and will require a regulator; the normalization denominator does not cancel it because the diagram remains connected to $x$ and $y$.

3. Count loop momenta before integrating

In quartic scalar theory, consider the connected four-point diagram with two vertices joined by two internal lines and with two external legs on each vertex. Find $L$ and write a consistent pair of internal momenta in terms of one loop momentum and the external momenta.

Solution

The graph has $I=2$ internal lines and $V=2$ vertices, so

$$L=I-V+1=1.$$

If $P=p_1+p_2$ enters the left vertex and all external momenta are oriented inward, choose one internal line to carry $\ell$ from left to right. Momentum conservation makes the other carry $P-\ell$ in the same left-to-right orientation. Reversing an internal orientation changes the sign assigned to that line’s momentum but not its scalar propagator. The unrenormalized loop integrand is proportional to

$$\int\frac{\mathrm d^d\ell}{(2\pi)^d} \frac{i}{\ell^2-m^2+i0} \frac{i}{(P-\ell)^2-m^2+i0},$$

times the two vertex factors and the graph’s symmetry factor. Regularization and interpretation belong to the loops lesson.

Continue to physical external states

You are ready to continue when you can derive the scalar propagator and vertex from the action, reproduce both factorials above by counting contractions, assign one independent momentum per loop, and explain the origin of each fermionic or gauge-theory sign you use.

Next, LSZ reduction and tree amplitudes removes the external propagators and connects the contact correlator to normalized scattering amplitudes and rates. If loop integrations are already present, do not evaluate them by an implicit prescription; continue afterward to Loops and regularization.

References

• Matthew D. Schwartz, Quantum Field Theory and the Standard Model, Cambridge University Press, 2014, doi:10.1017/9781139540940.
• Mark Srednicki, Quantum Field Theory, Cambridge University Press, 2007, doi:10.1017/CBO9780511813917.
• Steven Weinberg, The Quantum Theory of Fields, Volume I: Foundations, Cambridge University Press, 1995, doi:10.1017/CBO9781139644167.