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.

For the scalar-capstone calculation, the Gaussian background suffices for the scalar derivations below. The fermion and gauge extensions use the additional preparation just listed; they remain part of the full graduate core.

Split the action without changing the theory

Take a real scalar with $m>0$ and a weak real coupling $\lambda\ge0$, so the classical potential is bounded below, and write

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

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.

Use the same ultraviolet regulator and vacuum boundary prescription in every term. Vacuum projection, or its perturbative adiabatic and $i0$ prescription, selects the interacting vacuum continuously connected to the free one. The fields inside free-vacuum expectations below are interaction-picture fields. The formulas define an expansion order by order; they do not establish convergence of the series or existence of an interacting continuum measure.

With this prescription, the normalized 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 $(1/i)\delta/\delta J$ inserts a field: a raw derivative brings down $i\phi$ because the source enters as $+i\int J\phi$. Thus

$$G^{(n)}(x_1,\ldots,x_n) =\left.\left(\frac1i\right)^n \frac{\delta^nZ[J]} {\delta J(x_1)\cdots\delta J(x_n)}\right|_{J=0}.$$

Differentiation of the Gaussian performs all Wick pairings. This is the algebra behind the diagrams.

Vacuum normalization cancels a specific contraction class

The first-order two-point function makes the cancellation explicit. Use a translation-invariant regulated free propagator and restrict the vertex integrals to the same finite spacetime region until the cancellation. At one quartic vertex $z$, the six fields in the numerator have $5!!=15$ complete pairings. They split into two classes:

• Pair $x$ with $y$ and pair the four vertex fields among themselves in three ways. This is a free external propagator times a vacuum component.
• Pair $x$ and $y$ with distinct vertex fields in $4\cdot3=12$ ways, then pair the remaining two fields. This is the attached tadpole.

With $D_F(0)$ denoting the regulated coincidence limit, define

$$\begin{aligned} B_1&=-\frac{i\lambda}{8}\int\mathrm d^d z\,D_F(0)^2, \\ T_1(x,y)&=-\frac{i\lambda}{2}\int\mathrm d^d z\, D_F(x-z)D_F(0)D_F(z-y). \end{aligned}$$

The coefficients are $3/4!=1/8$ and $12/4!=1/2$. The numerator and denominator of the vacuum ratio are therefore

$$\begin{aligned} \text{numerator}&=D_F(x-y)(1+B_1)+T_1(x,y)+O(\lambda^2), \\ \text{denominator}&=1+B_1+O(\lambda^2). \end{aligned}$$

Multiplying by the inverse denominator $1-B_1+O(\lambda^2)$ gives

$$G^{(2)}(x,y)=D_F(x-y)+T_1(x,y)+O(\lambda^2).$$

The vacuum component cancels before any regulator limit. The tadpole survives because it connects to the insertions; its ultraviolet behavior must still be treated. Vacuum-amplitude normalization removes the disconnected vacuum components as described in Weinberg 1995, § 9.2, p. 389. At higher orders, disconnected vacuum components exponentiate into the same common factor in numerator and denominator. Follow the repeated-component factorial in Connected, Disconnected, and Vacuum Diagrams to see why this first-order cancellation extends to every perturbative order.

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 2006 draft, § 9, p. 76, and § 10, pp. 88–91, PDF 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

$$\frac{i(p\!\!\!/+m)}{p^2-m^2+i0}, \qquad p\!\!\!/\equiv\gamma^\mu p_\mu.$$

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 Faddeev–Popov determinant or equivalent BRST treatment. In QED with a linear covariant gauge, that determinant is independent of the gauge field, so the ghosts decouple even when charged matter interacts. Non-Abelian gauges generally require ghost vertices and loops. Gauge-dependent propagators and vertices are allowed intermediate objects; physical quantities must satisfy the appropriate Ward or Slavnov–Taylor identities. Compare these cases in Gauge-Fixed and Ghost Feynman Rules.

For an on-shell external photon and physical on-shell external states, replacing its polarization by its momentum should give zero after all relevant diagrams are summed. At tree level this gives the check

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

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 graph has symmetry factor $S=2$, hence coefficient $1/S=1/2$. 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. Determine a one-loop four-point contribution

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. For the fixed channel $(p_1,p_2)|(p_3,p_4)$, find $L$, determine the numerical coefficient by Wick counting, and write the complete amputated integrand with its overall momentum delta function suppressed.

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.

There are two assignments of the labeled external pairs to the two integration vertices. At each vertex the two labeled external legs attach in $4\cdot3$ ways; the remaining two slots at one vertex pair with those at the other in $2!$ ways. Thus the numerical coefficient is

$$\frac{2(4\cdot3)^2\,2!}{2!(4!)^2}=\frac12.$$

The denominator contains the Dyson $2!$ and the two interaction $4!$ factors. Independently, the two parallel internal lines may be exchanged while all external labels stay fixed, so $S=2$. The unrenormalized amputated contribution for this channel is

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

with a common regulator understood. The other two external pair partitions give the other channels; they are separate contributions, not an extra symmetry factor. Evaluating and renormalizing the integral belongs 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 the contact, tadpole, and fixed-channel loop coefficients by counting contractions, explain which first-order term the vacuum denominator cancels, and assign one independent momentum per loop. If a factor of $i$ or a pairing count is uncertain, return to the solved Gaussian source exercise in Functional integrals and correlators and then repeat these three checks. Before using the fermionic or gauge branches, also explain the origin of each extra sign and physical-state condition you introduce.

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, prepublication draft, 2006. Author’s source and errata page; Open PDF.
• Steven Weinberg, The Quantum Theory of Fields, Volume I: Foundations, Cambridge University Press, 1995, doi:10.1017/CBO9781139644167.