4.6 Feynman rules for real scalar QFTs

Our calculations in the previous sections may seem a bit cumbersome at first sight, but they all follow the same pattern which can be summarized by a small number of Feynman rules. Using such Feynman rules, one can avoid redoing again and again the same type of calculations that are needed to determine from the Gell-Mann and Low formula (4.16) and Wick’s theorem (4.32) the interacting time-ordered \(n\)-point functions and the scattering amplitudes of a QFT. To generalize a bit what was written in the previous sections, let us consider here the case of \(\Phi ^N\)-theory for an arbitrary \(N\geq 2\), i.e. we work with the interaction term

\begin{flalign} S_{\mathrm {int}}[\Phi _0] = -\frac {\lambda }{N!} \int _{\bbR ^d} \big (\Phi _0(x)\big )^N\,\dd x\quad . \end{flalign}

Time-ordered \(n\)-point functions:

The interacting time-ordered \(n\)-point function

\begin{flalign} \expect {\Omega }{\TO \big (\Phi (x_1)\cdots \Phi (x_n)\big )}{\Omega } \end{flalign} at order \(\lambda ^k\) can be computed by carrying out the following steps:

• • Draw \(n\) external points \(x_1,\dots , x_n\) and \(k\) internal points \(y_1, y_2,\dots , y_k\). The latter represent the \(k\) interaction vertices arising at order \(\lambda ^k\).

• • Connect the points by lines such that

  • – \(1\) line is attached to each external point \(x_i\),

  • – \(N\) lines are attached to each internal point \(y_j\).

  In general, there will be different ways in which one can connect the points, leading to different Feynman diagrams.

• • Record all (topologically distinct) Feynman diagrams and discard those including vacuum bubbles, see Observation 4.6.

• • Associate to each Feynman diagram an analytical expression by assigning

  • – to each external point \(x_i\) a factor \(1\),

  • – to each internal point \(y_j\) a factor \(-\ii \,\lambda \int _{\bbR ^d} \dd y_j\),

  • – to each line between two points \(z_a\) and \(z_b\) (these can be external or internal) a free Feynman propagator \(\Delta _F(z_a-z_b)\).

  These factors and Feynman propagators are then multiplied and weighted by the symmetry factor of the diagram. Note that the ordering of points in the Feynman propagators does not matter for a real scalar field because \(\Delta _F(z_a-z_b) = \Delta _F(z_b-z_a)\).

• • Sum up the individual contributions associated with distinct Feynman diagrams.

It would be a good exercise for you to redo our calculations in Section 4.3 by using directly the Feynman rules.

Scattering amplitudes:

Computing \(n\to m\) scattering amplitudes

\begin{flalign} \label {eqn:ntomscatteringKG} \braket {q_1,\dots ,q_m;\mathrm {out}}{k_1,\dots ,k_n;\mathrm {in}} \end{flalign} is related to computing time-ordered \(n+m\)-point functions, but there are some slight differences due to the prefactors in the LSZ formula (4.70) which have the effect of “amputating external legs”. As we have already seen in Section 4.5, only those Feynman diagrams in which each external point is connected to an internal point survive this amputation since the zeros in the prefactors of the LSZ formula turn all other contributions to zero when going on-shell. When considering higher orders in the coupling constant, it is also important to note that the amputation described by the LSZ formula (4.70) involves the interacting mass \(m_{\mathrm {int}}^2\) and (the square root of) the wave function renormalization \(Z\), hence it takes care of the loop corrections the particles receive while propagating to or from the interaction region. This reduces further the number of Feynman diagrams that contribute to the scattering amplitude. To understand which Feynman diagrams we should take into account, let us note that any Feynman diagram can be factorized as

The big blob in the middle describes the actual particle interaction and the small blobs on the external legs describe the loop corrections received by the particles while propagating to or from the interaction region. Since the latter are the source of the interacting mass \(m_{\mathrm {int}}^2\) and the wave function renormalization \(Z\), they will be amputated by the prefactors in the LSZ formula (4.70), leading to

\begin{flalign} \label {eqn:LSZamputated} \braket {q_1,\dots ,q_m;\mathrm {out}}{k_1,\dots ,k_n;\mathrm {in}}\, = \, (\sqrt {Z})^{n+m}~~\widetilde {G}_{n+m}\big (-k_1,\dots ,-k_n,q_1,\dots ,q_m\big )\Big \vert _{\mathrm {amputated}}^{}\quad . \end{flalign} Hence, only those Feynman diagrams with a trivial blob on each external leg have to be considered for computing scattering amplitudes. For example, for \(2\to 2\) scattering in \(\Phi ^3\)-theory, diagrams such as

have to be considered, while diagrams such as

do not have to be considered. It is also important to note the prefactor \(\sqrt {Z}^{n+m}\) in the amputated LSZ formula (4.85), which arises because the residue of each interacting Feynman propagator is by definition \(Z\) (see Observation 4.8) while the LSZ formula (4.70) only divides by \(1/\sqrt {Z}\) for each external leg. As a consequence, one has to know the corrections to \(Z=1+\mathcal {O}(\lambda )\), which are determined by computing the interacting Feynman propagator, in order to compute higher order corrections to scattering amplitudes. Luckily, if one is only interested in scattering amplitudes at leading order in the coupling constant \(\lambda \), as we were in our examples in Section 4.5, one can safely set \(Z=1\).

With these preparations, we can now write down the Feynman rules for scattering amplitudes, which we will do immediately in Fourier space since this is practically more convenient. The \(n\to m\) scattering amplitude (4.83) with \(k\) interaction vertices can be computed by carrying out the following steps:

• • Draw \(n\) external points on the left, \(m\) external points on the right and \(k\) internal points in the middle. These points represent, respectively, the incoming particles, the outgoing particles and the interaction vertices.

• • In analogy to the case of time-ordered correlation functions, connect these points by lines to produce Feynman diagrams (excluding vacuum bubbles). Due to the amputation of external legs, consider only those Feynman diagrams in which 1.) each external point is connected by a line to an internal point and 2.) the blob on each external leg is trivial.

• • Decorate the lines of these Feynman diagrams with relativistic momenta by assigning

  • – the incoming momenta \(k_a\) (pointing towards the interaction region) to the incoming external legs,

  • – the outgoing momenta \(q_b\) (pointing away from the interaction region) to the outgoing external legs,

  • – an arbitrary momentum \(l_c\) (pointing in the direction you like) to each internal line \(c\).

• • Associate to each decorated Feynman diagram an analytical expression by assigning

  • – to each (incoming or outgoing) external leg a factor of \(\sqrt {Z}\),

  • – to each internal line with momentum \(l_c\) a free Feynman propagator \(\frac {-\ii }{l_c^2 +m^2 -\ii \,\epsilon }\),

  • – to each interaction vertex a factor \(-\ii \,\lambda \,(2\pi )^d\, \delta \big (p_\mathrm {in} -p_{\mathrm {out}}\big )\) with \(p_{\mathrm {in}}\) the total incoming momentum at the vertex and \(p_\mathrm {out}\) the total outgoing momentum at the vertex. (This enforces relativistic momentum conservation at each vertex.)

  These factors and Feynman propagators are then multiplied, weighted by the symmetry factor of the diagram and integrated over all internal momenta \(\prod _c \int _{\bbR ^d}\frac {\dd l_c}{(2\pi )^d}\).

• • Sum up the individual contributions associated with distinct Feynman diagrams.

You might be surprised by the “momentum decoration rules” for Feynman diagrams, which is something we haven’t explicitly used in our (more rudimentary) diagrammatics in Section 4.5. The reason for this is that all diagrams in Section 4.5 are so simple that it is immediately obvious how to assign a momentum to each line by enforcing relativistic momentum conservation at all vertices. This won’t be the case anymore for more complicated diagrams, which is why the “momentum decoration rules” are a useful concept.

To practice the Feynman rules for scattering amplitudes, let us redo in detail the first diagram in (4.79). (This diagram arises for \(2\to 2\) scattering in \(\Phi ^3\)-theory.) We redraw this diagram with momentum decoration

which when evaluated with the Feynman rules gives

\begin{flalign} \nn &Z^2\,\int _{\bbR ^d} (-\ii \, \lambda ) \, (2\pi )^{d}\,\delta \big (k_1 + k_2 - l\big )~\frac {-\ii }{l^2 +m^2 -\ii \,\epsilon }~ (-\ii \, \lambda ) \, (2\pi )^{d}\,\delta \big (l - q_1 -q_2\big )~~\frac {\dd l}{(2\pi )^d}\\ &\qquad \qquad = Z^2\, (-\ii \,\lambda )^2\, (2\pi )^d\, \delta (k_1+k_2 - q_1-q_2) ~\frac {-\ii }{ (k_1+k_2)^2 + m^2 -\ii \,\epsilon }\quad . \end{flalign} Using that \(Z = 1+\mathcal {O}(\lambda )\) and sending \(\epsilon \to 0\), we find the same result as previously in the first term of (4.80). The other diagrams in (4.79) can be treated similarly, which allows us to recover the result (4.80) by using Feynman rules.

Going to higher orders in the coupling constant, there are of course many more Feynman diagrams one has to consider. It is a nontrivial task to write all of them down in a systematic fashion, which is why QFT practitioners are typically using computer assistance to ensure that they do not forget diagrams. To see some examples of such higher order Feynman diagrams for scattering amplitudes, let us consider again \(2\to 2\) scattering in \(\Phi ^3\)-theory. Among the many diagrams that contribute at order \(\lambda ^4\), we have the following examples

A common feature of all these diagrams is that they contain closed loops, which in many cases will cause divergences as previously seen in Warning 4.5. These divergences will be treated later in Chapter 8 when we introduce renormalization techniques. To practice once more the application of Feynman rules, let us evaluate the “box diagram” in the bottom left corner: First of all, we add relativistic momentum decorations to this diagram

Applying the Feynman rules then gives the expression

\begin{flalign} \nn &\sqrt {Z}^4\,\int _{\bbR ^{4d}}\frac {-i}{l_1^2 +m^2-\ii \,\epsilon } ~ \frac {-i}{l_2^2 +m^2-\ii \,\epsilon } ~ \frac {-i}{l_3^2 +m^2-\ii \,\epsilon } ~ \frac {-i}{l_4^2 +m^2-\ii \,\epsilon }~\times \\[5pt] \nn & \qquad \qquad (-\ii \,\lambda )\,(2\pi )^d\,\delta (k_1-l_1-l_2)~ (-\ii \,\lambda )\,(2\pi )^d\,\delta (k_2+l_2-l_3)~\times \\[5pt] \nn & \qquad \qquad (-\ii \,\lambda )\,(2\pi )^d\,\delta (l_3+l_4-q_2)~ (-\ii \,\lambda )\,(2\pi )^d\,\delta (l_1- q_1-l_4)~\times \\[5pt] & \qquad \qquad \frac {\dd l_1}{(2\pi )^d}~\frac {\dd l_2}{(2\pi )^d}~\frac {\dd l_3}{(2\pi )^d}~\frac {\dd l_4}{(2\pi )^d}\quad . \label {eqn:tmplongFeynman} \end{flalign} Carrying out the momentum space integrals over \(l_2,l_3,l_4\) by using that “delta functions kill integrations”, one obtains the following simplified expression

\begin{flalign} \nn & Z^2\, (-\ii \,\lambda )^4\, (2\pi )^d\, \delta (k_1+k_2-q_1-q_2)\, \int _{\bbR ^d} \frac {-\ii }{l^2 + m^2 -\ii \,\epsilon }~\frac {-\ii }{(l-q_1)^2 + m^2 -\ii \,\epsilon }~\times \\ &\qquad \qquad \qquad \frac {-\ii }{(k_1-l)^2 + m^2 -\ii \,\epsilon }~ \frac {-\ii }{(k_1+k_2-l)^2 + m^2 -\ii \,\epsilon }~\frac {\dd l}{(2\pi )^d}\quad .\label {eqn:tmpshortFeynman} \end{flalign} There is again a Dirac delta function that enforces overall relativistic momentum conservation for the external legs (\(k_1 + k_2 = q_1 + q_2\)), which is a common feature of all scattering amplitudes that is linked to the Poincaré invariance of our QFTs. The momenta entering the four Feynman propagators in (4.92) have been determined by using relativistic momentum conservation at each interaction vertex, which is enforced by the delta functions associated to vertices in (4.91). There are different (but equivalent) ways how to evaluate the integrals over these delta functions, and our choice can be visualized as follows

Further reading

For more details about the interacting quantum Klein-Gordon field in the operator approach, see e.g. Nastase (Chapters 5 and 6), Greiner/Reinhardt (Chapters 8 and 9) and Maggiore (Chapter 5) from our reading list in Section 1.3.