4.5 Simple examples of scattering amplitudes

To conclude this chapter, we shall work out some examples of scattering amplitudes to the lowest order in the coupling constants. We shall focus on \(2\to 2\) scatting amplitudes \(\braket {q_1,q_2;\mathrm {out}}{k_1,k_2;\mathrm {in}}\), which means that there are two incoming particles with relativistic momenta \(k_1,k_2\in \bbR ^d\) that interact with each other and produce two outgoing particles with different relativistic momenta \(q_1,q_2\in \bbR ^d\). In pictures, this means that we consider scatterings of the following type

where, as we shall see below, the grey blob admits a description in terms of Feynman diagrams.

Since our current goal is to describe scattering amplitudes to the lowest nontrivial order in the coupling constants, we can set in the LSZ formula (4.68), and in its Fourier space version (4.70), the wave function renormalization \(Z=1\) to be one and the interacting mass \(m^2_{\mathrm {int}}=m^2\) to be the parameter from the classical action functional. This is consistent because the corrections to \(Z\) and \(m^2_{\mathrm {int}}\) are of higher order in the coupling constants, see Observations 4.7 and 4.8. (It is however important to emphasize that such corrections to \(Z\) and \(m^2_{\mathrm {int}}\) are important for computing consistently the higher order corrections to scattering amplitudes.) This reduces the problem to computing the time-ordered \(4\)-point function

\begin{flalign} \expect {\Omega }{\TO \big (\Phi (x_1)\Phi (x_2)\Phi (y_1)\Phi (y_2)\big )}{\Omega } \end{flalign} or, in view of the LSZ formula in Fourier space (4.70), its Fourier transform

\begin{flalign} \nn \widetilde {G}_4(-k_1,-k_2,q_1,q_2)&= \int _{\bbR ^{4d}} e^{\ii \,(k_1 x_1 + k_2 x_2 - q_1 y_1 - q_2 y_2)}~\times \\ &\qquad \qquad \expect {\Omega }{\TO \Big (\Phi (x_1)\Phi (x_2)\Phi (y_1)\Phi (y_2)\Big )}{\Omega }~\dd x_1\, \dd x_2\, \dd y_1\, \dd y_2\quad . \end{flalign}

Let us start by analyzing the problem at order \(\lambda ^0\) in the coupling constants (i.e. for a free QFT) in order to get some feeling for the zero vs. pole issue from Warning 4.9. In this case the interacting time-ordered \(4\)-point function coincides with the free one. Using also Example 4.4, we obtain

It is important to note that there is no interaction vertex in the third diagram; the two crossing lines are a graphical representation of the product of Feynman propagators \((\Delta _F)_{x_1y_2} \,(\Delta _F)_{x_2 y_1}\). Passing over to Fourier space, we obtain

\begin{flalign} \nn \widetilde {G}_4(-k_1,-k_2,q_1,q_2) &= (2\pi )^{2d}\, \bigg (\frac {-\ii }{k_1^2 + m^2 -\ii \,\epsilon } ~ \frac {-\ii }{q_1^2 + m^2 -\ii \,\epsilon }\,\delta (k_1+k_2)\,\delta (q_1+q_2) \\ \nn &\qquad + \frac {-\ii }{k_1^2 + m^2 -\ii \,\epsilon } ~ \frac {-\ii }{k_2^2 + m^2 -\ii \,\epsilon }\,\delta (k_1-q_1)\,\delta (k_2-q_2) \\ &\qquad + \frac {-\ii }{k_1^2 + m^2 -\ii \,\epsilon } ~ \frac {-\ii }{k_2^2 + m^2 -\ii \,\epsilon }\,\delta (k_1-q_2)\,\delta (k_2-q_1) \bigg ) + \mathcal {O}(\lambda ^1)\quad , \end{flalign} where we note that the Dirac delta functions enforce the conservation of the relativistic momentum along the lines drawn in our Feynman diagrams. Inserting this into the LSZ formula in Fourier space (4.70) however gives zero, because the two poles of \(\widetilde {G}_4(-k_1,-k_2,q_1,q_2)\) are not sufficient to compensate the four zeros in its prefactor. Hence, we find that the scattering amplitude

\begin{flalign} \braket {q_1,q_2;\mathrm {out}}{k_1,k_2;\mathrm {in}} = 0 +\mathcal {O}(\lambda ^1) \end{flalign} vanishes in the case of no interactions, which is of course totally sensible since scattering should happen only when the particles interact with each other.

In order to get some nontrivial scattering amplitudes, we shall again study two examples of interaction terms, namely \(\Phi ^4\)-theory and \(\Phi ^3\)-theory as in Section 4.3.

\(\Phi ^4\)-theory:

Using Wick’s theorem (4.32) and the Observation 4.6 that vacuum bubbles do not contribute to the computation of the interacting time-ordered \(n\)-point functions, one finds that the interacting time-ordered \(4\)-point function of \(\Phi ^4\)-theory to first order in \(\lambda \) is given by

Passing over to Fourier space and inserting the result into the LSZ formula (4.70), we observe that only the last diagram contributes to the scattering amplitude because all other terms do not have sufficiently many poles to compensate the four zeros in the prefactor of (4.70). Performing a routine Fourier transform calculation, one finds

\begin{flalign} \label {eqn:2to2Phi4tree} \braket {q_1,q_2;\mathrm {out}}{k_1,k_2;\mathrm {in}} = -\ii \,\lambda \,(2\pi )^d\,\delta (k_1 + k_2 - q_1 - q_2) ~+~\mathcal {O}(\lambda ^2)\quad . \end{flalign} Note again that there is a Dirac delta function that enforces relativistic momentum conservation, i.e. the total incoming momentum \(k_1 +k_2\) must be the same as the total outgoing momentum \(q_1 + q_2\) for the scattering to take place. Furthermore, the coupling constant \(\lambda \), which controls the interaction term in the action functional \(S_{\mathrm {int}}[\Phi ] = -\frac {\lambda }{4!} \int _{\bbR ^d}\Phi ^4(x)\,\dd x\), determines the magnitude of the scattering amplitude.

\(\Phi ^3\)-theory:

Using again Wick’s theorem (4.32) and the Observation 4.6 that vacuum bubbles do not contribute to the computation of the interacting time-ordered \(n\)-point functions, one finds that the interacting time-ordered \(4\)-point function of \(\Phi ^3\)-theory to second order in \(\lambda \) is given by

It is a good exercise for you to determine these additional diagrams and try to understand why they do not contribute to the scattering amplitude (4.70). Performing once more a routine Fourier transform calculation, one finds

\begin{flalign} \nn &\braket {q_1,q_2;\mathrm {out}}{k_1,k_2;\mathrm {in}} = (-\ii \,\lambda )^2 \,(2\pi )^d\,\delta (k_1 + k_2 - q_1 - q_2)~\times \\ &\qquad ~\qquad \bigg (\frac {-\ii }{(k_1+k_2)^2 + m^2} + \frac {-\ii }{(k_1-q_1)^2 + m^2} + \frac {-\ii }{(k_1-q_2)^2 + m^2}\bigg ) ~+~\mathcal {O}(\lambda ^3)\quad . \label {eqn:scatteringPhi3value} \end{flalign} Note that the internal lines of Feynman diagrams contribute with a Feynman propagator (in Fourier space), whose momentum is determined by relativistic momentum conservation at each interaction vertex. These internal lines are interpreted as virtual particles, because their relativistic momenta do not necessarily satisfy the on-shell conditions, i.e. \((k_1+k_2)^2 \neq - m^2\), \((k_1-q_1)^2 \neq - m^2\) and \((k_1-q_2)^2 \neq - m^2\). Such virtual particles are a quantum mechanical phenomenon that does not have any analog in classical physics. The mechanism that interactions between particles are mediated by virtual particles is very common in QFT and it also appears in physically relevant examples, such as quantum electrodynamics (QED) or the standard model of particle physics.