3 Interacting Quantum Fields
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The theories we have looked at so far are described by Gaussian partition functions or, equivalently, actions that are quadratic in the fields. These theories are trivial, in the sense that all \(N\)-point functions (which collectively describe everything there is to know) decompose identically into \(2\)-point functions. To do useful calculations we need to consider interacting theories. The particles can either interact with themselves, or we can have multiple types of particle which can interact with one another.
3.1 \(\varphi^{4}\) theory
We will focus on the first case: a scalar field with self-interactions, called \(\varphi^{4}\) theory. The action is
\[S=S_{0}+\int\text{d}^{D}x\varphi_{x}J_{x}+S_{\text{int}} \tag{3.1}\]
where
\[\begin{aligned} S_{0} & \triangleq S_{\text{KG}}\\ S_{\text{int}} & \triangleq-\frac{\lambda}{4!}\int\text{d}^{D}x\varphi^{4}_{x} \end{aligned}\]
with \(\lambda>0\). The minus sign is required because the Lagrange density is kinetic minus potential energy: the interaction adds a potential energy density \(+\lambda\varphi^{4}/4!\), which is bounded below, so the theory has a stable ground state.
You may hazard a guess as to where the name comes from! This action does not describe any particles in the standard model (nor does Klein Gordon), but it does describe other physical theories. For example, phonons emerging from balls and springs with an anharmonic potential between atoms, or the Ising model of interacting spins. The normalisation \(1/4!\) turns out to be convenient later.
3.2 Perturbation Theory
To calculate the \(N\)-point functions for a non-Gaussian theory we can use perturbation theory. The basic idea is pretty simple. The generating functional is now given by
\[\begin{aligned} \mathscr{Z}_{\lambda}\left[J\right] & =\int\mathscr{D}\varphi\exp\left(iS_{0}-i\frac{\lambda}{4!}\int\text{d}^{D}x\varphi^{4}_{x}+i\int\text{d}^{D}x\varphi_{x}J_{x}\right)\\ & =\int\mathscr{D}\varphi\exp\left(iS_{0}+i\int\text{d}^{D}x\varphi_{x}J_{x}\right)\exp\left(-i\frac{\lambda}{4!}\int\text{d}^{D}x\varphi^{4}_{x}\right) \end{aligned}\]
and we can expand the exponential of the interaction term as a Taylor series to give
\[\mathscr{Z}_{\lambda}\left[J\right]=\int\mathscr{D}\varphi\sum^{\infty}_{n=0}\frac{1}{n!}\left(-i\frac{\lambda}{4!}\int\text{d}^{D}x\varphi^{4}_{x}\right)^{n}\exp\left(iS_{0}+i\int\text{d}^{D}x\varphi_{x}J_{x}\right).\]
At each order in \(\lambda\) this is simply the expectation value of a polynomial of the fields, and any such object can be calculated using Wick’s theorem. Let’s take a look at some specific examples.
3.2.1 The interacting vacuum
The non-interacting field theory had an empty vacuum. But when interactions are present, the vacuum changes its form. To see this, we can calculate the 0-point function1, which is simply \(\mathscr{Z}_{\lambda}\left[0\right]\) itself (the generating functional with the source switched off). To \(\mathcal{O}\left(\lambda^{2}\right)\) we have:
\[\mathscr{Z}_{\lambda}\left[0\right]=\int\mathscr{D}\varphi\left(1-i\frac{\lambda}{4!}\int\text{d}^{D}x\varphi^{4}_{x}+\frac{1}{2}\left(-i\frac{\lambda}{4!}\right)^{2}\int\text{d}^{D}x\int\text{d}^{D}y\varphi^{4}_{x}\varphi^{4}_{y}\right)\exp\left(iS_{0}\right).\]
We work order-by-order in \(\lambda\), and find
\[\frac{\mathscr{Z}_{\lambda}\left[0\right]}{\mathscr{Z}_{0}\left[0\right]}=1-i\frac{\lambda}{4!}\int\text{d}^{D}x\langle\varphi^{4}_{x}\rangle_{0}+\frac{1}{2}\left(-i\frac{\lambda}{4!}\right)^{2}\int\text{d}^{D}x\int\text{d}^{D}y\langle\varphi^{4}_{x}\varphi^{4}_{y}\rangle_{0}+\mathcal{O}\left(\lambda^{3}\right)\]
where we have introduced the convenient notation
\[\left\langle A\right\rangle _{0}\triangleq\frac{\int\mathscr{D}\varphi A\exp\left(iS_{0}\right)}{\int\mathscr{D}\varphi\exp\left(iS_{0}\right)}\]
i.e. the familiar expectation value without the interaction term. We proceed using Wick’s theorem, which says that any such \(\left\langle A\right\rangle _{0}\) expectation value is equal to the sum of all possible products of 2-point functions. This gives
\[\begin{aligned} \frac{\mathscr{Z}_{\lambda}\left[0\right]}{\mathscr{Z}_{0}\left[0\right]} & =1-i\frac{\lambda}{4!}\int\text{d}^{D}x3\langle\varphi^{2}_{x}\rangle_{0}\langle\varphi^{2}_{x}\rangle_{0}\\ & +\frac{1}{2}\left(-i\frac{\lambda}{4!}\right)^{2}\int\text{d}^{D}x\int\text{d}^{D}y\left\{ 3\times3\langle\varphi^{2}_{x}\rangle_{0}\langle\varphi^{2}_{x}\rangle_{0}\langle\varphi^{2}_{y}\rangle_{0}\langle\varphi^{2}_{y}\rangle_{0}\right.\\ & +\left.\phantom{}^{4}C_{2}\times\phantom{}^{4}C_{2}\times2\langle\varphi_{x}\varphi_{y}\rangle_{0}\langle\varphi_{x}\varphi_{y}\rangle_{0}\langle\varphi^{2}_{x}\rangle_{0}\langle\varphi^{2}_{y}\rangle_{0}+4!\langle\varphi_{x}\varphi_{y}\rangle^{4}_{0}\right\} \\ & +\mathcal{O}\left(\lambda^{3}\right). \end{aligned}\]
The counting is as follows. In the \(\mathcal{O}\left(\lambda\right)\) term, there are three ways for the first \(\varphi_{x}\) to pick a partner, and then the other pair is fixed. Similarly for the first \(\mathcal{O}\left(\lambda^{2}\right)\) term (for \(x\) and \(y\) separately). For the second \(\mathcal{O}\left(\lambda^{2}\right)\) term there are \(\phantom{}^{4}C_{2}\) ways to choose one \(xx\) pair; same for the \(yy\) pair; then there are two ways to choose the first \(xy\) pair and the second is fixed.
We now use the fact that the non-interacting 2-point function is the Green’s function to give
\[\begin{aligned} \frac{\mathscr{Z}_{\lambda}\left[0\right]}{\mathscr{Z}_{0}\left[0\right]} & =1-i\lambda\int\text{d}^{D}x\frac{1}{8}G^{2}_{xx}\\ & +\left(-i\lambda\right)^{2}\int\text{d}^{D}x\int\text{d}^{D}y\left\{ \frac{1}{2\cdot8^{2}}G^{2}_{xx}G^{2}_{yy}+\frac{1}{16}G_{xx}G^{2}_{xy}G_{yy}+\frac{1}{2\cdot4!}G^{4}_{xy}\right\} \\ & +\mathcal{O}\left(\lambda^{3}\right). \end{aligned}\]
Each term is a vacuum bubble: a diagram with no external points (we will draw them in Fig. 3.2). Their sum is just a number, \(\mathscr{Z}_{\lambda}\left[0\right]/\mathscr{Z}_{0}\left[0\right]\), and as we will now see it cancels whenever we compute a properly normalised correlation function.
3.2.2 Interacting 2-point function
Next we can redo the calculation of the 2-point function from Section 2.9. Correlation functions are normalised by the full \(\mathscr{Z}_{\lambda}\left[0\right]\):
\[\begin{aligned} \langle\mathcal{T}\varphi_{2}\varphi_{1}\rangle & =\left.\mathscr{Z}^{-1}_{\lambda}\left[0\right]\left(-i\frac{\delta}{\delta J_{1}}\right)\left(-i\frac{\delta}{\delta J_{2}}\right)\mathscr{Z}_{\lambda}\left[J\right]\right|_{J=0}\\ & =\frac{\int\mathscr{D}\varphi\,\varphi_{2}\varphi_{1}\left(1-i\frac{\lambda}{4!}\int\text{d}^{D}x\varphi^{4}_{x}+\ldots\right)\exp\left(iS_{0}\right)}{\int\mathscr{D}\varphi\left(1-i\frac{\lambda}{4!}\int\text{d}^{D}x\varphi^{4}_{x}+\ldots\right)\exp\left(iS_{0}\right)}\\ & =\langle\mathcal{T}\varphi_{2}\varphi_{1}\rangle_{0}-i\frac{\lambda}{4!}\int\text{d}^{D}x\left(\left\langle \mathcal{T}\varphi^{4}_{x}\varphi_{2}\varphi_{1}\right\rangle _{0}-\left\langle \mathcal{T}\varphi_{2}\varphi_{1}\right\rangle _{0}\left\langle \varphi^{4}_{x}\right\rangle _{0}\right)+\mathcal{O}\left(\lambda^{2}\right) \end{aligned}\]
where in the last line we expanded the denominator, \(\left(1-\epsilon\right)^{-1}\approx1+\epsilon\). The \(\mathcal{O}\left(\lambda^{0}\right)\) term is again the non-interacting expression. Let’s evaluate the first \(\mathcal{O}\left(\lambda\right)\) term, which comes from the numerator. All possible contractions are shown schematically in Fig. 3.1.
The result, counting all possibilities, is
\[\begin{aligned} -i\frac{\lambda}{4!}\int\text{d}^{D}x\left\langle \mathcal{T}\varphi^{4}_{x}\varphi_{1}\varphi_{2}\right\rangle _{0} & =-i\frac{\lambda}{4!}\int\text{d}^{D}x\left\{ 3\left\langle \varphi^{2}_{x}\right\rangle _{0}\left\langle \varphi^{2}_{x}\right\rangle _{0}\left\langle \varphi_{1}\varphi_{2}\right\rangle _{0}+4\times3\left\langle \varphi_{x}\varphi_{1}\right\rangle _{0}\left\langle \varphi_{x}\varphi_{2}\right\rangle _{0}\left\langle \varphi^{2}_{x}\right\rangle _{0}\right\} \\ & =-i\lambda\int\text{d}^{D}x\left\{ \frac{1}{8}G^{2}_{xx}G_{12}+\frac{1}{2}G_{x1}G_{x2}G_{xx}\right\} . \end{aligned} \tag{3.2}\]
The first term is the free propagator multiplied by a vacuum bubble, and it is exactly cancelled by the term coming from the denominator (since \(\langle\varphi^{4}_{x}\rangle_{0}=3G^{2}_{xx}\)). This is the general pattern: normalising by \(\mathscr{Z}_{\lambda}\left[0\right]\) removes all diagrams containing vacuum bubbles. We are left with
\[\langle\mathcal{T}\varphi_{1}\varphi_{2}\rangle=G_{12}-i\frac{\lambda}{2}\int\text{d}^{D}xG_{1x}G_{xx}G_{x2}+\mathcal{O}\left(\lambda^{2}\right). \tag{3.3}\]
The counting of terms is as follows. For the \(G^{2}_{xx}G_{12}\) term, the first \(x\) has \(3\) ways to choose the second \(x\) with which it contracts. The remaining two \(x\)s are then forced. Hence, \(3\). For the \(G_{x1}G_{x2}G_{xx}\) term, there are \(4\) choices of \(x\) for the \(1\) to pair with, leaving \(3\) choices for the \(2\) to pair with, then the remaining pair of \(x\)s must go together. Hence \(4\times3\).
3.2.3 Interacting 4-point function
Since the 3-point function vanishes, the next simplest term is the 4-point function. At order \(\lambda^{0}\) this is again the same as the unperturbed case. At order \(\lambda^{1}\) the numerator gives
\[\begin{aligned} -i\frac{\lambda}{4!}\int\text{d}^{D}x\left\langle \mathcal{T}\varphi^{4}_{x}\varphi_{1}\varphi_{2}\varphi_{3}\varphi_{4}\right\rangle _{0} & =-i\frac{\lambda}{4!}\int\text{d}^{D}x\left\{ 4!G_{1x}G_{2x}G_{3x}G_{4x}+3G^{2}_{xx}\left(G_{12}G_{34}+G_{13}G_{24}+G_{14}G_{23}\right)\right.\\ & +\left.4\times3G_{xx}\left(G_{1x}G_{2x}G_{34}+G_{1x}G_{3x}G_{24}+G_{1x}G_{4x}G_{23}+G_{2x}G_{3x}G_{14}+G_{2x}G_{4x}G_{13}+G_{3x}G_{4x}G_{12}\right)\right\} \\ & =-i\lambda\int\text{d}^{D}x\left\{ G_{1x}G_{2x}G_{3x}G_{4x}+\frac{1}{8}G^{2}_{xx}\left(G_{12}G_{34}+2\text{ perms}\right)+\frac{1}{2}G_{xx}\left(G_{1x}G_{2x}G_{34}+5\text{ perms}\right)\right\} . \end{aligned} \tag{3.4}\]
As before, dividing by \(\mathscr{Z}_{\lambda}\left[0\right]\) removes the terms containing the vacuum bubble \(G^{2}_{xx}\). The remaining terms containing \(G_{xx}\) are not vacuum bubbles, but they are disconnected: they describe the \(\mathcal{O}\left(\lambda\right)\) correction to one particle’s propagator, multiplied by the free propagator of a second particle which never interacts with the first. Only the first term describes the two particles scattering off one another.
At higher orders of \(\lambda\) things quickly grow in combinatorial difficulty. There is a neat bookkeeping trick for dealing with this, to which we turn now.
3.3 Feynman Diagrams
3.3.1 Diagrams
Feynman invented his eponymous notation to simplify the rapidly growing complexity of the combinatorics just discussed.
The Feynman diagrams for the expansions in the previous sections appear in Fig. 3.2 (the vacuum), Fig. 3.3 (2-point function), and Fig. 3.4 (4-point function).
Take the 2-point function. You start by drawing a vertex for each external field position (anything not integrated over, or equivalently anything explicitly labelled on the left): in our case, \(x_{1}\) and \(x_{2}\). Then draw a single point for each internal field position, in our case \(x\). Now find all the ways to draw lines connecting these points, so that each external point meets precisely one line, and each internal point meets precisely 4 lines.
The reason each internal point has 4 lines meeting is that this is \(\varphi^{4}\) theory. When you Taylor expand the action, every power of \(\lambda\) brings with it a product \(\varphi^{4}_{x}\), so there is no other possibility for lines to meet. E.g. you can’t have 2 lines meet at an internal point.
3.3.2 Symmetry factors
A remarkable result of the Feynman diagrams is that they instantly yield the difficult combinatorial pre-factors. We term the ‘symmetry factor’ of a Feynman diagram the number of ways its internal lines and vertices can be interchanged while leaving the diagram (with its external points held fixed) unchanged. The prefactor of each term in the perturbative expansion is then given by the reciprocal of the diagram’s symmetry factor. To see why this is, it’s easiest to look at some examples.
In Eq. 3.2 for the 2-point functions to order \(\lambda\), the first term is \(\frac{1}{8}G^{2}_{xx}G_{12}\). Looking at the corresponding diagram in Fig. 3.3 we see that the \(G_{12}\) line is fixed (\(x_{1}\) and \(x_{2}\) are fixed), but the \(G^{2}_{xx}\) has some freedom. It has two lobes. We can flip the left lobe (\(\times2\)), the right lobe (\(\times2\)), and we can interchange the lobes (\(\times2\)). The symmetry factor is \(2\times2\times2=8\), and so the prefactor is \(1/8\).
The second term is \(\frac{1}{2}G_{x1}G_{x2}G_{xx}\). The corresponding diagram has one internal loop, which can be flipped (\(\times2\)). Hence the diagram has a symmetry factor of 2, and the prefactor is \(1/2\).
Now look at the 4-point functions in Eq. 3.4, and the corresponding Feynman diagrams in Fig. 3.4. The term \(G_{1x}G_{2x}G_{3x}G_{4x}\) has a prefactor of 1. This is actually by construction: it is why we included the \(1/4!\) factor in the definition of \(S_{\text{int}}\). There are 4 ways for \(1\) to choose an \(x\), 3 ways for the \(2\), 2 ways for the \(3\), and one way for the 4, giving \(4!\) terms, cancelling the prefactor. Looking at the diagram, all legs connect to external events, so there is no freedom.
Other symmetry factors can be worked out using similar reasoning. Often it is quite tricky, in the general case; but practical QFTs such as QED are more restricted and typically have simpler symmetry factors.
3.3.3 Physical interpretation of Feynman diagrams
Feynman diagrams suggest a natural physical interpretation. Initially, it can be helpful to draw space and time axes (e.g. time heading up, space horizontal). Then you can interpret the lines (propagators) as worldlines of particles. Heading up, if lines meet (which must happen at a vertex), the particles annihilate. If lines emerge from a vertex, particles are created.
In \(\varphi^{4}\) theory each vertex must be the meeting point of 4 lines, so if two head in, then two head out, and this is like two particles annihilating and another two being created at the same event. The spacetime points of internal vertices are always integrated over: this means we create a quantum superposition of that event happening at any spacetime location.
Consider the 2-point function connecting events \(x_{1}\) and \(x_{2}\), which gives the probability amplitude to detect a particle at \(x_{1}\) and at \(x_{2}\) (recalling that this doesn’t necessarily mean the same particle travels between those events – as the amplitude is non-zero for spacelike separation). In non-interacting \(\varphi^{4}\) theory, there is an amplitude for the particle to propagate from \(x_{1}\) to \(x_{2}\). In the interacting theory, this is also possible. But inspecting Fig. 3.3 (rightmost diagram), we see there is also an amplitude for the particle to start propagating from \(x_{1}\), then to emit a particle at \(x\), to catch that same particle also at \(x\), then to propagate to \(x_{2}\).
Giving a causal interpretation to intermediate events in Feynman diagrams is therefore tricky. Closed loops appear, and along any closed loop any spacetime axes must suggest a particle propagating forwards and backwards in time. That’s how the intermediate particle can depart from \(x\) then arrive at \(x\).
The spacetime axes are arbitrary: you could have drawn space heading up, time horizontal, and you would have found a different physical interpretation. This is a helpful reminder that the only measurable things occur at spacetime events corresponding to external vertices.
3.3.4 Feynman Rules
The Feynman diagrams are in one-to-one correspondence with the terms in the perturbation expansion of an interacting QFT. They are typically easy to draw, and it’s fairly easy to check they’ve all been included at a given order. Hence, QFT typically amounts to drawing Feynman diagrams first, then converting them into their analytic expressions. This conversion is carried out by applying ‘Feynman rules’.
For the case above, we have:
The Feynman rules for \(\varphi^{4}\) theory (position space):
To each propagator (line) assign \(G_{xy}\)
To each vertex assign \(-i\lambda\int\text{d}^{D}x\)
Divide by the symmetry factor.
In high-energy physics it is more common to work with the Fourier transformed propagators, since the boundary conditions typically take the form of specifying the momentum of a particle before and after a collision (recalling that the position and momentum cannot be simultaneously specified). In this case we have
The Feynman rules for \(\varphi^{4}\) theory (momentum space):
To each propagator (line) assign \(\tilde{G}_{p}\)
To each vertex assign \(-i\lambda\left(2\pi\right)^{D}\delta^{D}\left(p_{1}+p_{2}+p_{3}+p_{4}\right)\) (momentum conservation, counting all momenta as flowing into the vertex)
integrate over each internal momentum with \(\int\text{d}^{D}k/\left(2\pi\right)^{D}\)
Divide by the symmetry factor.
3.3.5 The Connected Generating Functional
Inspecting and interpreting the Feynman diagrams, we see that many of them are rather trivial. There are two kinds. Diagrams containing vacuum bubbles, such as \(G_{12}G^{2}_{xx}\) (the free propagator along with an event occurring elsewhere in spacetime, at \(x\), connected to nothing), are removed by normalising by \(\mathscr{Z}_{\lambda}\left[0\right]\), as we saw above: the vacuum is precisely the sum of all bubbles. But even then some diagrams are disconnected, like the \(G_{xx}G_{1x}G_{2x}G_{34}\) terms in the 4-point function: two particles propagating without ever interacting. Since fields were introduced to act locally, and we are usually interested in how particles interact, what we are really interested in is connected diagrams: those in which the whole diagram is one connected piece.
There is a very elegant trick for calculating only connected diagrams. Motivated by the observation that
\[\frac{\text{d}}{\text{d}x}\ln f=f^{-1}\frac{\text{d}f}{\text{d}x}\]
and the desire to divide out the vacuum, we replace our generating functional \(\mathscr{Z}\left[J\right]\) with
The connected generating functional:
\[\mathscr{W}_{\lambda}\left[J\right]\triangleq\ln\left(\mathscr{Z}_{\lambda}\left[J\right]\right).\]
We use it precisely as we used \(\mathscr{Z}_{\lambda}\left[J\right]\), but now we find only connected diagrams! For example, consider the 2-point function to \(\mathcal{O}\left(\lambda\right)\). We have \[\mathscr{W}_{\lambda}\left[J\right]=\ln\int\mathscr{D}\varphi\sum^{\infty}_{n=0}\frac{1}{n!}\left(-i\frac{\lambda}{4!}\int\text{d}^{D}x\varphi^{4}_{x}\right)^{n}\exp\left(iS_{0}+i\int\text{d}^{D}x\varphi_{x}J_{x}\right)\]
which to \(\mathcal{O}\left(\lambda\right)\) is
\[\begin{aligned} \mathscr{W}_{\lambda}\left[J\right] & =\ln\left[\mathscr{Z}_{0}\left[J\right]-i\frac{\lambda}{4!}\int\mathscr{D}\varphi\left(\int\text{d}^{D}x\varphi^{4}_{x}\right)\exp\left(iS_{0}+i\int\text{d}^{D}x\varphi_{x}J_{x}\right)\right]\\ & =\ln\left(\mathscr{Z}_{0}\left[J\right]\right)+\ln\left[1-i\frac{\lambda}{4!}\mathscr{Z}^{-1}_{0}\left[J\right]\int\mathscr{D}\varphi\left(\int\text{d}^{D}x\varphi^{4}_{x}\right)\exp\left(iS_{0}+i\int\text{d}^{D}x\varphi_{x}J_{x}\right)\right] \end{aligned}\]
which can itself be Taylor expanded in \(\lambda\), using
\[\ln\left(1+\lambda\epsilon\right)=\lambda\epsilon+\mathcal{O}\left(\lambda^{2}\right)\]
to give
\[\mathscr{W}_{\lambda}\left[J\right]=\ln\left(\mathscr{Z}_{0}\left[J\right]\right)-i\frac{\lambda}{4!}\mathscr{Z}^{-1}_{0}\left[J\right]\int\mathscr{D}\varphi\left(\int\text{d}^{D}x\varphi^{4}_{x}\right)\exp\left(iS_{0}+i\int\text{d}^{D}x\varphi_{x}J_{x}\right).\]
After a reasonable bit of algebra (make sure to drop any terms you can spot will be zero along the way, e.g. \(\langle\varphi_{1}\varphi^{4}_{x}\rangle_{0}\)) you will find for the connected 2-point function
\[\begin{aligned} \langle\mathcal{T}\varphi_{1}\varphi_{2}\rangle_{\text{connected}} & =\left.\left(-i\frac{\delta}{\delta J_{2}}\right)\left(-i\frac{\delta}{\delta J_{1}}\right)\mathscr{W}_{\lambda}\left[J\right]\right|_{J=0}\\ & =\langle\mathcal{T}\varphi_{1}\varphi_{2}\rangle_{0}-i\frac{\lambda}{4!}\int\text{d}^{D}x\left(\langle\mathcal{T}\varphi_{1}\varphi_{2}\varphi^{4}_{x}\rangle_{0}-\langle\mathcal{T}\varphi_{1}\varphi_{2}\rangle_{0}\langle\mathcal{T}\varphi^{4}_{x}\rangle_{0}\right). \end{aligned}\]
The second term in parentheses subtracts off the vacuum bubble. At higher orders, and for higher \(N\)-point functions, taking the logarithm subtracts off every disconnected piece too. The connected generating functional similarly leads to connected diagrams at all orders, and for all correlation functions.
3.4 Regularisation and Renormalisation
The point of all this perturbation theory is to find how interactions alter the ‘bare’, non-interacting \(N\)-point functions. Broadly, the answer is that the bare functions become renormalised. There are unfortunately two essentially totally unrelated meanings of the word ‘renormalised’, both introduced in QFT, and both of which are relevant here! Let’s look at them in turn.
3.4.1 Renormalisation (Meaning I)
One of the main things we’d like to know about is the interacting 2-point function. The non-interacting 2-point function is the bare propagator for our theory. Adding interactions ‘dresses’ the propagator, changing its form. Often we will be interested to see what happens to one type of particle when we introduce a second type of particle with which it can interact. Typically we will expect that interactions will ‘renormalise’ the properties of both particles, meaning for example that their effective masses or charges change.
A simple picture to hold in mind is to imagine trying to measure the mass of a tennis ball: you can do this by trying to accelerate the ball with a tennis racket, and measuring how much force it takes to impart a given acceleration. Now try the same experiment under water. It will take more force to impart that same acceleration, since the tennis ball is now interacting with the water molecules. Hence, your experiment tells you that the tennis ball’s mass has been renormalised. This analogy also captures the fact that the renormalisation is accompanied by a change in the nature of the ‘vacuum’ itself, from air to water.
This idea applies wherever QFT is used. For example, the bare electron propagator in the standard model corresponds to a particle whose mass does not match the measured mass of the electron, because the electrons you can measure have already been renormalised by interactions with other fields (notably photons, but also \(W\) bosons, Higgs bosons, and anything that couples to electromagnetism or mass).
Often in condensed matter physics we are interested in precisely this type of scenario: two fields, with their associated particle types, where the particles of one field move much faster (have much larger characteristic energies) than those of the other. In such cases, we typically carry out the functional integral over the fast field using perturbation theory, leading to a renormalised effective theory of the slow one. We call this process ‘integrating out’ a field, and it’s a vital procedure in QFT. Let’s look at a real example now.
3.4.2 The random phase approximation (Renormalisation I)
Consider the interaction of fast electrons with slow phonons in a crystal. Here, we have the QFT
\[\begin{aligned} S & =S_{0}+S_{\text{int}}\\ S_{0} & =\frac{1}{2}\int\text{d}^{D}q\varphi_{q}D^{-1}_{q}\varphi_{-q}+\int\text{d}^{D}k\psi^{\dagger}_{k}G^{-1}_{k}\psi_{k}\\ S_{\text{int}} & =g\int\text{d}^{D}q\int\text{d}^{D}k\varphi_{q}\psi^{\dagger}_{k}\psi_{k+q} \end{aligned}\]
where \(k=\left(\vb{k},\omega\right)\), and \(\int\text{d}^{D}k\) is shorthand for a sum over \(\vb{k}\) and an integral over \(\omega\) (we suppress factors of volume and \(2\pi\), and write the propagators at real frequency: a careful treatment works at imaginary, Matsubara, frequencies and continues back at the end),
\[D_{q}=\frac{-2\Omega_{\vb{q}}}{\omega^{2}-\Omega^{2}_{\vb{q}}}\]
is the bare phonon propagator for phonons with energy \(\Omega_{\vb{q}}\) at wavevector \(\vb{q}\), and
\[G_{k}=\frac{1}{\omega-\xi_{\vb{k}}}\]
is the bare electron propagator, where \(\xi_{\vb{k}}\) is the energy of an electron with crystal momentum \(\vb{k}\). The energy of a particle is given by the locations of the poles in its Green’s function. The positive energy pole of the bare phonon is therefore
\[\omega=\Omega_{\vb{q}}.\]
We can look at how the interactions with the electron field change this. If you try drawing connected Feynman diagrams to correct the phonon propagator, you will find that the lowest order diagram appears at order \(g^{2}\). It is shown in Fig. 3.5. However, there is a neat trick by which we can include an infinite number of Feynman diagrams of a similar form, also shown in that figure. We call this the random phase approximation or RPA (the name is for historical, and rather unconvincing, reasons). The fermions here are described by Grassmann fields, which we meet in Chapter 4; the only consequence we need now is that every closed fermion loop comes with a factor of \(-1\).
It is important to note that the set of diagrams in the RPA is not all possible diagrams renormalising the phonon propagator. Rather, it is a mathematically convenient subset to consider, and in general there is no guarantee that the discarded connected diagrams (the first of which appears at order \(g^{4}\)) are small. RPA is therefore in general an uncontrolled approximation: there is no small parameter which controls the error. (It does become controlled in certain limits, for example when there are many species of fermion, or for the long-range Coulomb interaction at high density.) This caveat in mind, let’s proceed to calculate the RPA corrected phonon energy.
The loop which appears inside the propagator is called the polarisation bubble. Including the factor of \(-1\) for the closed fermion loop, it contributes \(-g^{2}\chi_{q}\), where
\[\chi_{q}=\int\text{d}^{D}kG_{k}G_{k+q}.\]
The frequency integral can be carried out using a contour integral, with the result (per spin species)
\[\chi_{q}=\frac{1}{V}\sum_{\vb{k}}\frac{f\left(\xi_{\vb{k}}\right)-f\left(\xi_{\vb{k}+\vb{q}}\right)}{\omega+\xi_{\vb{k}}-\xi_{\vb{k}+\vb{q}}+i0^{+}}\]
which is the Lindhard function, where
\[f\left(\xi\right)=\frac{1}{\exp\left(\frac{\xi}{T}\right)+1}\]
is the Fermi Dirac distribution. The wavevectors are discrete since the crystal is finite, exactly as in Chapter 1. The sum requires \(\xi_{\vb{k}}\) to be specified in order to be evaluated, and this is generally done numerically. Note that the static Lindhard function is negative: at \(\omega=0\), whenever \(\xi_{\vb{k}+\vb{q}}>\xi_{\vb{k}}\) we have \(f\left(\xi_{\vb{k}}\right)>f\left(\xi_{\vb{k}+\vb{q}}\right)\), so each term is negative. The interacting phonon propagator is then
\[D^{\text{RPA}}_{q}=D_{q}+D_{q}\left(-g^{2}\chi_{q}\right)D_{q}+D_{q}\left(-g^{2}\chi_{q}\right)D_{q}\left(-g^{2}\chi_{q}\right)D_{q}+\ldots\]
This general form is called a Dyson series. It is simply a geometric series of Feynman diagrams. As such, the infinite sum can be carried out exactly, by noticing that
\[\begin{aligned} D^{\text{RPA}}_{q} & =D_{q}+D_{q}\left(-g^{2}\chi_{q}\right)\left(D_{q}+D_{q}\left(-g^{2}\chi_{q}\right)D_{q}+\ldots\right)\\ & =D_{q}-D_{q}g^{2}\chi_{q}D^{\text{RPA}}_{q} \end{aligned}\]
and so, subtracting the second term and gathering things together:
\[\left(1+D_{q}g^{2}\chi_{q}\right)D^{\text{RPA}}_{q}=D_{q}\]
giving the final result
\[\begin{aligned} D^{\text{RPA}}_{q} & =\left(1+D_{q}g^{2}\chi_{q}\right)^{-1}D_{q}\\ & =\frac{-2\Omega_{\vb{q}}}{\omega^{2}-\Omega^{2}_{\vb{q}}-2g^{2}\Omega_{\vb{q}}\chi_{q}}. \end{aligned}\]
This is shown diagrammatically in Fig. 3.6.
The positive pole of this renormalised propagator occurs at
\[\omega=\sqrt{\Omega^{2}_{\vb{q}}+2g^{2}\Omega_{\vb{q}}\chi_{q}}.\]
Since the (static) Lindhard function is negative, by interacting with the electrons the energy \(\omega\) of the phonon at a given wavevector has decreased: the phonons are softened by the electrons screening the ionic motion. (This neglects the frequency dependence of \(\chi_{q}\), which is reasonable since the phonons are slow compared to the electrons.)
3.4.3 1PI Diagrams and Self-Energies
The Dyson series used in the RPA diagram summation is so useful that it has become a standard method of organising Feynman diagrams in QFT. We define a ‘1-Particle Irreducible’ (1PI) diagram to be:
1PI diagram: any Feynman diagram which remains connected after cutting any one internal line.
A connected diagram is one in which, placing your pen on any line in the diagram, all points can be reached without removing your pen from the page.
An internal line is any line which connects only to internal vertices (those which are integrated over).
With this definition, we can always decompose any Feynman diagram into 1PI diagrams connected with single lines, and we can similarly decompose the infinite set of Feynman diagrams into a Dyson series. How do we know this is always possible? It is by definition! The Dyson series is defined to be the individual 1PI parts connected together with single edges (cutting any one of which will disconnect the diagram). So the Dyson series captures everything that is not 1PI, and since we’re summing everything, the bits in the 1PI blobs have to be all the bits of diagrams which remain connected after a single cut. You can think of 1PI diagrams as a bit like the prime numbers of Feynman diagrams: just as any integer can be factored into a product of primes, any Feynman diagram can be factored into a product of 1PI diagrams.
As in the RPA, we can now deduce the renormalisation of our propagator (or any other \(N\)-point function) by an infinite Dyson series of 1PI diagrams. When renormalising a propagator we define the 1PI blob to be a ‘self energy’ \(\Sigma\). Then, as in RPA, we can argue that
\[\begin{aligned} G^{\text{ren}} & =G+G\Sigma G^{\text{ren}}\\ & \downarrow\nonumber \\ G^{\text{ren}} & =\left(1-G\Sigma\right)^{-1}G. \end{aligned}\]
Example: non-relativistic electron propagator
The clearest example of this effect is with the (spinless, non-relativistic) electron propagator which is sometimes used in condensed matter physics. Here we have
\[G=\frac{1}{\omega-\xi_{\vb{k}}+i\epsilon}\]
where \(\epsilon\) is an infinitesimal positive real number (specifying how the pole shifts off the real axis in contour integrals; this choice gives the retarded Green’s function). Ignoring \(\epsilon\), the Green’s function has a pole at \(\omega=\xi_{\vb{k}}\), showing that the energy \(\omega\) of the electron with crystal momentum \(\vb{k}\) is \(\xi_{\vb{k}}\).
The probability to find the electron with wavevector \(\vb{k}\) and energy \(\omega\) is given by the ‘spectral function’
\[\begin{aligned} A\left(\vb{k},\omega\right) & \triangleq-\frac{1}{\pi}\mathfrak{Im}G\left(\vb{k},\omega\right)\\ & =\frac{1}{\pi}\frac{\epsilon}{\left(\omega-\xi_{\vb{k}}\right)^{2}+\epsilon^{2}}. \end{aligned}\]
Noting that this is simply a Lorentzian of width \(\epsilon\) we see that
\[\lim_{\epsilon\rightarrow0^{+}}A\left(\vb{k},\omega\right)=\delta\left(\omega-\xi_{\vb{k}}\right)\]
which says that the electron has 100% probability to be found with energy \(\omega=\xi_{\vb{k}}\). As a quick check that this makes sense, we can confirm that the electron must exist at some energy, and so
\[\int\text{d}\omega A\left(\vb{k},\omega\right)=1\]
which you can confirm by doing the Lorentzian integral. (This is called a ‘sum rule’). What happens when the electron propagator is renormalised by an infinite Dyson series? We find
\[\begin{aligned} G^{\text{ren}} & =\left(1-G\Sigma\right)^{-1}G\\ & =\frac{1}{\omega-\xi_{\vb{k}}-\Sigma+i\epsilon}. \end{aligned}\]
Writing
\[\Sigma\triangleq\Sigma'+i\Sigma''\]
for real \(\Sigma',\)\(\Sigma''\), the spectral function becomes
\[\begin{aligned} A\left(\vb{k},\omega\right) & =-\frac{1}{\pi}\mathfrak{Im}G\left(\vb{k},\omega\right)\\ & =\frac{1}{\pi}\frac{\epsilon-\Sigma''}{\left(\omega-\xi_{\vb{k}}-\Sigma'\right)^{2}+\left(\epsilon-\Sigma''\right)^{2}}. \end{aligned}\]
For the retarded Green’s function, causality requires \(\Sigma''\le0\) (a positive \(\Sigma''\) would describe a state whose amplitude grows exponentially in time), so \(A\ge0\) as a probability density must be. We can now drop \(\epsilon\ll\left|\Sigma''\right|\) to give
\[A\left(\vb{k},\omega\right)=\frac{1}{\pi}\frac{\left|\Sigma''\right|}{\left(\omega-\xi_{\vb{k}}-\Sigma'\right)^{2}+\Sigma''^{2}}.\]
Two things have happened. First, the real part of the self energy \(\Sigma'\) has shifted the peak of \(A\) to \(\omega=\xi_{\vb{k}}+\Sigma'\), renormalising the energy. Second, the imaginary part \(\Sigma''\) has given a finite width to the spectral function Lorentzian. The electron of wavevector \(\vb{k}\) is no longer certain to be at the peak in \(A\); rather, it can now have any energy (with the probability for it to have energy between \(\omega\) and \(\omega+\text{d}\omega\) given by \(A\text{d}\omega\)). Note that the sum rule is still obeyed: the total probability to find the electron with some energy is still \(1\). (In general \(\Sigma\) depends on \(\vb{k}\) and \(\omega\), so \(A\) is only approximately Lorentzian near its peak; the sum rule nevertheless holds exactly, as it follows from the anticommutation relations of the electron operators rather than from the shape of \(A\).)
3.4.4 Renormalisation II
In \(\varphi^{4}\) theory, the field interacts with itself. This self-interaction will renormalise (I) the \(N\)-point functions. To order \(\lambda\), with connected diagrams, the propagator is renormalised as (Eq. 3.3)
\[\begin{aligned} G_{12} & \rightarrow G_{12}-i\frac{\lambda}{2}\int\text{d}^{D}xG_{1x}G_{xx}G_{x2}. \end{aligned}\]
This time, however, there is a problem. The term
\[G_{xx}=G\left(x-x\right)=G\left(0\right)=\int\frac{\text{d}^{D}p}{\left(2\pi\right)^{D}}\frac{i}{p^{2}-m^{2}+i0^{+}}\]
is infinite. (It is the size of the free field’s fluctuations at a single point, \(\langle\varphi^{2}_{x}\rangle_{0}\).) We saw similar infinities before, in the zero-point energy of Section 1.7.4, which we eliminated by requiring operators to be normal ordered. The analogy is close: \(G_{xx}\) arises from contracting two fields at the same point, and if we had normal ordered the interaction, \(:\varphi^{4}:\), this particular diagram would be removed. But the general problem is not removed by normal ordering, so let’s understand it properly. This type of infinity is called an ‘ultraviolet divergence’, since it has occurred in the high energy / high momentum / small distance limit \(x-x=0\). (Ultraviolet is used by analogy to light, where UV is high energy compared to the visible spectrum; similarly Infrared, or IR, is used to describe long wavelength / low momentum / low energy properties). To deal with UV divergences, we use a method called... renormalisation! But despite appearing at the same time in these notes, the two uses are essentially unrelated. Let’s look at what I’ll call Renormalisation II. First, let’s write out the correction explicitly using the expression for the Green’s function
\[G_{xy}=\int\frac{\text{d}^{D}p}{\left(2\pi\right)^{D}}\frac{i}{p^{2}-m^{2}}\exp\left(-ip\cdot\left(x-y\right)\right)\]
to give (the \(x\) integral sets the two remaining momenta equal)
\[-i\frac{\lambda}{2}\int\text{d}^{D}xG_{1x}G_{xx}G_{x2}=i\frac{\lambda}{2}G\left(0\right)\int\frac{\text{d}^{D}q}{\left(2\pi\right)^{D}}\frac{\exp\left(-iq\cdot\left(x_{1}-x_{2}\right)\right)}{\left(q^{2}-m^{2}\right)^{2}}.\]
The remaining \(q\) integral is fine; the problem is \(G\left(0\right)\). To get an idea of how such integrals behave, we can use a simple dimensional analysis. Roughly, we expect an integral of the form
\[\int^{\infty}\text{d}^{D}pp^{n}\]
to be convergent (at large \(p\)) for \(n<-D\) and divergent for \(n>-D\). For \(n=-D\) we expect to get a logarithm, which will diverge with infinite limits (log-divergent: the slowest type of divergence). \(G\left(0\right)\) has \(n=-2\), so is divergent for \(D\ge2\).
The first step to dealing with it is regularisation. The integrals diverge because of the infinite limits. But in reality, momenta cannot be infinitely large, and distances cannot be infinitely small. In an extreme limit, distances cannot be smaller than the Planck length \(l_{\text{P}}=\sqrt{G}\) (where \(G\) is Newton’s constant, in natural units). But we don’t need to go to such extremes: our field theory is a coarse-graining, and will only be valid down to some distance / up to some momentum (for the chain of Chapter 1, the lattice spacing \(a\) provides exactly such a cutoff, \(\Lambda\sim\pi/a\)). Call the upper momentum cutoff \(\Lambda\), which is vastly larger than any momentum we care about but is not infinite. Returning to our divergent integral
\[G\left(0\right)=\int\frac{\text{d}^{D}p}{\left(2\pi\right)^{D}}\frac{i}{p^{2}-m^{2}+i0^{+}}\]
it is convenient to Wick rotate to Euclidean space. The Feynman prescription allows the \(p^{0}\) contour to be rotated anticlockwise onto the imaginary axis without crossing any poles, so we can set \(p^{0}=ip^{0}_{\text{E}}\), with \(\text{d}^{D}p=i\text{d}^{D}p_{\text{E}}\) and \(p^{2}=-p^{2}_{\text{E}}\), where \(p^{2}_{\text{E}}=\left(p^{0}_{\text{E}}\right)^{2}+\vb{p}^{2}\) is a Euclidean square. This gives
\[G\left(0\right)=\int\frac{\text{d}^{D}p_{\text{E}}}{\left(2\pi\right)^{D}}\frac{1}{p^{2}_{\text{E}}+m^{2}}\]
(which is manifestly positive, as a variance must be). Specialising to \(D=4\), and using 4D spherical polar co-ordinates (the surface area of the unit 3-sphere is \(2\pi^{2}\)), gives
\[\begin{aligned} G\left(0\right)=\frac{2\pi^{2}}{\left(2\pi\right)^{D}}\int^{\Lambda}_{0}\text{d}p\frac{p^{3}}{p^{2}+m^{2}} & =\frac{1}{16\pi^{2}}\left[\Lambda^{2}-m^{2}\ln\left(1+\frac{\Lambda^{2}}{m^{2}}\right)\right]\\ & \approx\frac{1}{16\pi^{2}}\left[\Lambda^{2}-m^{2}\ln\left(\frac{\Lambda^{2}}{m^{2}}\right)\right] \end{aligned}\]
which contains a quadratic divergence and a log divergence.
What does this correction do? It is easiest to see in momentum space, where the propagator is \(\tilde{G}\left(p\right)=i/\left(p^{2}-m^{2}\right)\) and the tadpole is a 1PI self-energy insertion \(-i\lambda G\left(0\right)/2\). Summing the Dyson series as in the previous section,
\[\tilde{G}^{\text{ren}}\left(p\right)=\frac{\tilde{G}}{1-\tilde{G}\left(-i\lambda G\left(0\right)/2\right)}=\frac{i}{p^{2}-m^{2}-\frac{\lambda}{2}G\left(0\right)}.\]
The pole has moved: the physical mass is \(m^{2}_{\text{phys}}=m^{2}+\frac{\lambda}{2}G\left(0\right)\), which diverges as \(\Lambda\rightarrow\infty\). From here, we simply renormalise (II) by subtracting off this infinite amount. We do this by introducing a ‘counterterm’ in the Lagrange density:
\[\mathcal{L}=\frac{1}{2}\partial^{\mu}\varphi\partial_{\mu}\varphi-\frac{1}{2}m^{2}\varphi^{2}-\frac{1}{2}\delta_{m}\varphi^{2}-\frac{\lambda}{4!}\varphi^{4} \tag{3.5}\]
where \(\delta_{m}\) is a new parameter, of order \(\lambda\). We treat the counterterm as a new interaction: it gives a two-point vertex contributing \(-i\delta_{m}\), which enters the self-energy alongside the tadpole. The physical mass becomes
\[m^{2}_{\text{phys}}=m^{2}+\frac{\lambda}{2}G\left(0\right)+\delta_{m}.\]
We are free to choose \(\delta_{m}\) so that the parameter \(m\) in the Lagrangian is the physical mass we measure (this is called the ‘on-shell’ scheme):
\[\delta_{m}=-\frac{\lambda}{2}G\left(0\right)=-\frac{\lambda}{32\pi^{2}}\left[\Lambda^{2}-m^{2}\ln\left(\frac{\Lambda^{2}}{m^{2}}\right)\right].\]
Putting it all together, the corrected Lagrange density in Eq. 3.5 gives the propagator \(i/\left(p^{2}-m^{2}\right)\) to first order in \(\lambda\), with no divergence in sight. By construction, the UV divergence has been subtracted off, by renormalising the mass (through an infinite amount!). That is, with this choice, the propagator does not renormalise at 1st order. The reason we can do this with a simple constant is that \(G_{xx}\) is special, as it carries no dependence on the external momentum. Its Feynman diagram is called a ‘tadpole’ (when drawn correctly). Tadpoles can only ever shift the mass, and can be renormalised away for free. At higher orders, divergent diagrams do depend on the external momenta, and we also need counterterms for the normalisation of the field and for the coupling \(\lambda\); the process of identifying these self-consistently, and of tracking how the renormalised parameters depend on the scale at which they are measured, leads to the renormalisation group.
If we look at the philosophy of Renormalisation II, we see that it is secretly quite similar to Renormalisation I. The idea is that the bare parameters we started with – in this case \(m^{2}+\delta_{m}\) – are not the physical parameters. The things we measure have already been renormalised, and so it is \(m^{2}_{\text{phys}}\) which is physical. Still, the mathematical process and reasoning behind Renormalisation I and II appear rather different.
I’ve not seen it called this, but it seems a sensible name to me!↩︎