4.3 Interacting Feynman propagator

In order to get a better understanding of Wick’s theorem, let us compute the lowest order corrections to the interacting Feynman propagator

\begin{flalign} \label {eqn:interactingKGpropagator} \expect {\Omega }{\TO \big (\Phi (x_1)\,\Phi (x_2)\big )}{\Omega } = \frac {\expect {0}{\TO \Big (\Phi _0(x_1)\,\Phi _0(x_2) \, e^{\ii \,S_{\mathrm {int}}[\Phi _0]}\Big )}{0}}{\expect {0}{\TO \Big (e^{\ii \, S_{\mathrm {int}}[\Phi _0]}\Big )}{0}}\quad . \end{flalign} For this we have to choose an interaction term \(S_{\mathrm {int}}\) for the action functional. We consider the following two cases that are not only motivated by their simplicity, but also by their similarity to the kind of interactions that appear in the standard model of particle physics:

• • A cubic interaction term \(S_{\mathrm {int}}[\Phi _0] = -\frac {\lambda }{3!} \int _{\bbR ^d} \big (\Phi _0(x)\big )^3\,\dd x\), which is called \(\Phi ^3\)-theory.

• • A quartic interaction term \(S_{\mathrm {int}}[\Phi _0] = -\frac {\lambda }{4!} \int _{\bbR ^d} \big (\Phi _0(x)\big )^4\,\dd x\), which is called \(\Phi ^4\)-theory.

Of course, our formalism works for any interaction term \(S_{\mathrm {int}}[\Phi _0] =-\sum _{n\geq 3}\frac {\lambda _n}{n!} \int _{\bbR ^d} \big (\Phi _0(x)\big )^n\,\dd x\), but working at this level of generality would unnecessarily complicate our studies.

\(\Phi ^4\)-theory:

Let us start by looking at the numerator of (4.34) and expand it to the first order in the coupling constant \(\lambda \). Using a similar shorthand notation as in Section 4.2, we write \(\Phi _1 := \Phi _0(x_1)\), \(\Phi _2:= \Phi _0(x_2)\) and \(\Phi _y := \Phi _0(y)\). We obtain

\begin{flalign} \label {eqn:Phi4propagagtor} \expect {0}{\TO \Big (\Phi _1\,\Phi _2 \, e^{\ii \,S_{\mathrm {int}}[\Phi _0]}\Big )}{0} = \expect {0}{\TO \big (\Phi _1\,\Phi _2 \big )}{0} -\frac {\ii \,\lambda }{4!} \int _{\bbR ^d}\expect {0}{\TO \Big (\Phi _1\,\Phi _2\,(\Phi _y)^4 \Big )}{0}\,\dd y + \mathcal {O}(\lambda ^2)\quad . \end{flalign} We can evaluate these expressions by using Wick’s theorem, where the variant from Corollary 4.3 is particularly useful. The first term becomes a Feynman propagator \(\expect {0}{\TO \big (\Phi _1\,\Phi _2 \big )}{0} = (\Delta _F)_{12}\) and the second term yields a sum over all possible contractions of pairs according to (4.32). The latter contractions come in two different types:

• Type 1: \((\Delta _{F})_{12}\, (\Delta _F)_{yy}\,(\Delta _F)_{yy}\) is obtained by contracting \(\Phi _1\) with \(\Phi _2\) and the four \(\Phi _y\) among themselves.

• Type 2: \((\Delta _F)_{1y} \, (\Delta _F)_{2y}\,(\Delta _F)_{yy}\) is obtained by contracting \(\Phi _1\) with one of the \(\Phi _y\), \(\Phi _2\) with another one of the \(\Phi _y\), and the remaining two \(\Phi _y\) among themselves.

We can visualize these two types of contributions by the following Feynman diagrams that keep track of who is contracted against who:

The lines represent free Feynman propagators and the vertices represent the interaction term, which explains why they are \(4\)-valent for \(\Phi ^4\)-theory. The type 1 diagram can be interpreted physically as follows: While a particle propagates from \(x_1\) to \(x_2\), the vacuum undergoes a quantum fluctuation that creates and annihilates virtual particles. Such vacuum fluctuations that are not connected to any of the external points \(x_i\) are often called vacuum bubbles, which makes sense given the shape of the diagram. The interpretation of the type 2 diagram is as follows: A particle propagates from \(x_1\) and then transforms at some point \(y\) into three particles of which two annihilate each other and the third propagates to \(x_2\). This is also a quantum fluctuation, but in contrast to vacuum bubbles it is connected to the external points \(x_i\).

We still have to count the multiplicities of these two types of contributions, i.e. we have to figure out how often they appear in the sum (4.32). The associated combinatorial problem is to count in how many ways the set \(\{1,2,y,y,y,y\}\) can be partitioned into \(\{\{1,2\},\{y,y\},\{y,y\}\}\) (for type \(1\)) and into \(\{\{1,y\},\{2,y\},\{y,y\}\}\) (for type \(2\)).

• Type 1: There is \(1\) possibility to pick \(\{1,2\}\). Fixing any \(y\), there are \(3\) possibilities to pick its partner from the remaining three \(y\)’s to form a pair \(\{y,y\}\). The remaining pair \(\{y,y\}\) is fixed by these previous choices. Hence, there are \(3\) contributions to type \(1\).

• Type 2: There are \(4\) possibilities to pick a \(y\) for \(\{1,y\}\). Once this has been picked, we are left with \(3\) possibilities to pick a \(y\) for \(\{2,y\}\). The remaining pair \(\{y,y\}\) is fixed by these previous choices. Hence, there are \(4\times 3=12\) contributions to type \(2\).

With these preparations, we can now finally write down an expression for (4.35)

which we can also display graphically as

The prefactors in this graphical representation are called the symmetry factors and they are given by the inverse of the order of the symmetry group of the diagram that fixes the external points \(x_i\). In our example, the vacuum bubble diagram has a \(\bbZ _2\times \bbZ _2\times \bbZ _2\) symmetry given by 1.) flipping the left with the right bubble, 2.) flipping the two legs of the left bubble and 3.) flipping the two legs of the right bubble. This is a group of order \(2\times 2\times 2=8\), hence the symmetry factor is \(\frac {1}{8}\). The third diagram has only a \(\bbZ _2\)-symmetry given by flipping the two legs of the loop, hence the symmetry factor is \(\frac {1}{2}\).

By direct inspection, one sees that there is a simple way to translate between the graphical description and the explicit formula. This translation dictionary is given by the so-called Feynman rules, which for our example of \(\Phi ^4\)-theory read as follows:

• • To every line between two points \(x\) and \(y\) assign a Feynman propagator

  

• • To every vertex at a point \(z\) assign an integral

  

To complete the description of the interacting Feynman propagator (4.34) to first order in the coupling constant \(\lambda \), we still have to understand the denominator. Using the same methods as above, one finds in our graphical notation that

Applying now a first order Taylor expansion in \(\lambda \) to the ratio in (4.34), we find

You may now ask yourself: Is there any physical meaning and interpretation of the corrections (4.41) to the Feynman propagator? (At least qualitatively, because we haven’t yet renormalized the divergences that we have discovered in Warning 4.5.) There is indeed! Recall from (3.57) that the Fourier transform

\begin{flalign} \widetilde {\Delta }_F(k) \,:=\, \int _{\bbR ^d} \Delta _F(x)\, e^{-\ii \,k\,x}\,\dd x \,=\, \frac {-\ii }{k^2 + m^2 -\ii \,\epsilon } \end{flalign} of the free Feynman propagator has a pole at \(k^2=-m^2+\ii \,\epsilon \). Recall also that the parameter \(m^2\) from the classical action functional could be interpreted as the mass of the associated particles. Taking the Fourier transform of the interacting Feynman propagator (4.41), we find

\begin{flalign} \nn \widetilde {\Delta }^{\mathrm {int}}_F(k) \,&=\, \widetilde {\Delta }_F(k) -\frac {\ii \,\lambda }{2} \,\Delta _F(0)\,\Big (\widetilde {\Delta }_F(k)\Big )^2 + \mathcal {O}(\lambda ^2)\\ \nn \,&=\,\widetilde {\Delta }_F(k)~\bigg (1-\frac {\ii \,\lambda }{2} \,\Delta _F(0)\,\widetilde {\Delta }_F(k)\bigg ) + \mathcal {O}(\lambda ^2)\\ \nn \,&=\,\widetilde {\Delta }_F(k)~\sum _{n=0}^\infty \bigg (-\frac {\ii \,\lambda }{2} \,\Delta _F(0)\,\widetilde {\Delta }_F(k)\bigg )^n + \mathcal {O}(\lambda ^2)\\ \nn \,&=\,\frac {\widetilde {\Delta }_F(k)}{1+\frac {\ii \,\lambda }{2} \,\Delta _F(0)\,\widetilde {\Delta }_F(k)} \,+\, \mathcal {O}(\lambda ^2)\\ \,&=\,\frac {-\ii }{ k^2 + m^2 + \frac {\lambda }{2}\,\Delta _F(0) -\ii \,\epsilon }\,+\,\mathcal {O}(\lambda ^2)\quad , \label {eqn:KGPhi4massshift} \end{flalign} where in the step from the second to the third line we have extended the term in the parenthesis to a geometric series, which is of course the same up to \(\mathcal {O}(\lambda ^2)\).

\(\Phi ^3\)-theory:

Computing the interacting Feynman propagator (4.34) for \(\Phi ^3\)-theory works similarly as in the case of \(\Phi ^4\)-theory above. Hence, I will only spell out the relevant diagrams that contribute to this calculation and summarize the final result. I would recommend you to carry out these computations on your own to see if you get the details right.

Note that the first nontrivial correction to the Feynman propagator for \(\Phi ^3\)-theory is of order \(\lambda ^2\), which is due to the fact that all odd time-ordered \(n\)-point functions vanish by Corollary 4.3, hence in particular \(\expect {0}{\TO \big (\Phi _1\,\Phi _2\,\big (\Phi _y\big )^3\big )}{0}=0\). Recalling also from Observation 4.6 that all vacuum bubbles can be neglected, we find the following result

Note that the last term is the square of a contribution that already arises for the interacting \(1\)-point function

which describes the vacuum expectation value of the interacting field operator \(\Phi (x)\). Whenever the latter is nonzero, it makes sense to redefine the field operator \(\underline {\Phi }(x) := \Phi (x) - \expect {\Omega }{\Phi (x)}{\Omega }\) by subtracting its vacuum expectation value, which then describes the quantum fluctuation around \(\expect {\Omega }{\Phi (x)}{\Omega }\). At the level of the Feynman propagator, this subtraction amounts to considering the interacting connected Feynman propagator

It is worthwhile to note the similarity between the connected Feynman propagator and the concept of variance from probability theory. To translate this graphical description to an explicit formula, one can use the Feynman rules for \(\Phi ^3\)-theory:

• • To every line between two points \(x\) and \(y\) assign a Feynman propagator

  

• • To every vertex at a point \(z\) assign an integral

  

One then finds

\begin{flalign} \nn \expect {\Omega }{\TO \big (\Phi (x_1)\,\Phi (x_2)\big )}{\Omega }_{\mathrm {c}}^{} \,&=\, \Delta _{F}(x_1-x_2) -\frac {\lambda ^2}{2}\int _{\bbR ^{2d}} \Delta _F(x_1-y)\,\Delta _F(x_2-z)\,\big (\Delta _F(y-z)\big )^2\,\dd y\,\dd z\\ &-\frac {\lambda ^2}{2}\int _{\bbR ^{2d}} \Delta _F(x_1-y)\,\Delta _F(x_2-y)\,\Delta _F(y-z)\,\Delta _F(0) \,\dd y\,\dd z~+\, \mathcal {O}(\lambda ^3)\quad . \end{flalign} Having a closer look at this expression, one recognizes as in Warning 4.5 the appearance of divergences, whose renormalization will be discussed later.

To conclude this section, let us study as in (4.43) the Fourier transform of this expression and in particular its pole structure. Performing a routine Fourier transform calculation, one finds that

\begin{flalign} \nn \widetilde {\Delta }^{\mathrm {int}}_{F,c}(k) &= \widetilde {\Delta }_{F}(k) - \frac {\lambda ^2}{2} \big (\widetilde {\Delta }_F(k)\big )^2 \,\int _{\bbR ^d}\widetilde {\Delta }_F(q)\,\widetilde {\Delta }_F(k-q)\,\frac {\dd q}{(2\pi )^d}\\ \nn &\qquad -\frac {\lambda ^2}{2}\,\big (\widetilde {\Delta }_F(k)\big )^2\, \Delta _F(0)\,\widetilde {\Delta }_F(0) ~+~ \mathcal {O}(\lambda ^3)\\ &=\widetilde {\Delta }_{F}(k) ~\bigg (1 -\frac {\lambda ^2}{2}\,\widetilde {\Delta }_F(k) \Big (\mathcal {I}(k^2) +\mathcal {J}\Big ) \bigg )~+~ \mathcal {O}(\lambda ^3)\quad , \end{flalign} where in the last step have introduced the abbreviations \(\mathcal {I}(k^2):= \int _{\bbR ^d}\widetilde {\Delta }_F(q)\,\widetilde {\Delta }_F(k-q)\,\frac {\dd q}{(2\pi )^d}\) and \(\mathcal {J}:= \Delta _F(0)\,\widetilde {\Delta }_F(0)\). Note that the integral \(\mathcal {I}\) is Poincaré invariant, from which one can deduce that it has to be a function of \(k^2\). Using the same geometric series trick as in (4.43), we can rewrite this as

\begin{flalign} \widetilde {\Delta }^{\mathrm {int}}_{F,c}(k) =\frac {-\ii }{k^2 + m^2 -\ii \,\frac {\lambda ^2}{2}\,\big (\mathcal {I}(k^2) +\mathcal {J}\big ) - \ii \,\epsilon } ~+~\mathcal {O}(\lambda ^3)\quad . \end{flalign} The main difference between this result and (4.43) is that the correction term \(\mathcal {I}(k^2)\) may depend on the Fourier momentum. This has the following consequence: