8.2 Renormalized perturbation theory

In Observation 4.7, we have already seen indications that the bare parameters \(m_0^2\) and \(\lambda _0\) from the classical action (8.1) receive quantum corrections from loop diagrams and hence they do not describe the physical parameters of the QFT. Furthermore, as indicated in Observation 4.8, the field \(\Phi \) itself will in general receive quantum corrections to its normalization. Renormalized perturbation theory is a clever and practically very useful reformulation of the perturbation theory from Chapter 4 in which the action is expressed in terms of the physical/renormalized parameters, instead of the bare ones, and the renormalized field.

Let us explain how this works for \(\Phi ^4\)-theory in \(d=4\) spacetime dimensions, which is our prime example for a renormalizable QFT, see Example 8.5. Other examples of renormalizable QFTs, such as \(\Phi ^3\)-theory in \(d=6\) dimensions from Example 8.6, can be treated similarly. Let us first take care of the wave function renormalization by defining the renormalized field as

\begin{flalign} \Phi _r \,:= \,Z^{-\frac {1}{2}}\, \Phi \quad . \end{flalign} This rescaling is chosen to cancel the \(Z\)-factor in the interacting propagator, see Observation 4.8. (As a side-remark, note that this rescaling implies that, when working with \(\Phi _r\), there are no \(Z\)-factors in the LSZ formula (4.68).) Inserting \(\Phi = Z^{\frac {1}{2}}\,\Phi _r\) into the classical action (8.12) of \(d{=}4\)-dimensional \(\Phi ^4\)-theory, we obtain

\begin{flalign} S[\Phi _r] \,=\, \int _{\bbR ^4} -\bigg (\frac {Z}{2}\,\partial ^\mu \Phi _r\,\partial _\mu \Phi _r + \frac {m_0^2\,Z}{2}\,\Phi _r^2 + \frac {\lambda _0\,Z^2}{4!}\,\Phi _r^4\bigg )\,\dd x\quad . \end{flalign} The bare mass and coupling constant still appear in this action, but they can be eliminated as follows. Let us introduce

\begin{flalign} \label {eqn:counterterms} \delta _Z \,:=\, Z-1~~,\quad \delta _m \,:=\, m_0^2\,Z-m^2~~,\quad \delta _\lambda \,:=\, \lambda _0\,Z^2 - \lambda \quad , \end{flalign} where \(m^2\) and \(\lambda \) are the physically measured mass and coupling constant. Inserting this into the action, we obtain

The terms in the first line are the familiar ones from \(\Phi ^4\)-theory, but importantly they are expressed in terms of the physical/renormalized field \(\Phi _r\), mass \(m^2\) and coupling \(\lambda \). The terms in the second line are called the counterterms and, as we shall see later in this chapter, their role is to absorb the divergences in the renormalization process. So summing up, the advantage of working with the action in the form of (8.21) is that there is a clear split between the physical quantities, which must be finite, and the unphysical counterterms, which can be infinite due to divergences from loop diagrams.

The counterterms in the action (8.21) lead to additional Feynman rules on top of the ones from Section 4.6. For example, for scattering amplitudes in momentum space, the relevant Feynman rules are as follows:

• • Propagator

  

• • Interaction vertex

  

• • Propagator counterterm

  

• • Vertex counterterm

  

All of the above is meaningless until we state very clearly how the physical/renormalized parameters are linked to suitable quantum field theoretical quantities, such as time-ordered \(n\)-point functions or scattering amplitudes. It is important to emphasize that perturbative QFT is not capable to compute these parameters from first principles, but they rather have to be fixed via experimental input. Creating a link between the experimentally determined physical parameters and quantum field theoretical quantities is however still necessary in order to fix the so far undetermined counterterms \(\delta _Z\), \(\delta _m\) and \(\delta _\lambda \) in the action (8.21). In QFT jargon, such links are called renormalization conditions. We shall now discuss a particularly simple, but useful, set of renormalization conditions which is called the on-shell renormalization scheme. (There exist other renormalization schemes, e.g. the MS-scheme and MS-bar-scheme, but these will not be discussed in these lecture notes.) The idea behind the on-shell renormalization scheme is quite simple: To link our QFT to the experimentally determined physical mass \(m^2\), we consider the Fourier transform of the interacting Feynman propagator

\begin{flalign} \expect {\Omega }{\TO \big (\Phi _r(x)\,\Phi _r(y)\big )}{\Omega } = \int _{\bbR ^4}\widetilde {\Delta }^{\mathrm {int}}_F(k)\,e^{\ii \,k\,(x-y)}\,\frac {\dd k}{(2\pi )^4} \end{flalign} of the renormalized field \(\Phi _r\). (In the next section we will compute \(\widetilde {\Delta }^{\mathrm {int}}_F(k)\) at the \(1\)-loop level using our Feynman rules with counterterms from above.) The physical mass \(m^2\) is then identified as the pole of \(\widetilde {\Delta }^{\mathrm {int}}_F(k)\), i.e.

\begin{flalign} \label {eqn:interactingpropagatorRen} \widetilde {\Delta }^{\mathrm {int}}_F(k) \,=\, \frac {-\ii }{k^2+m^2-\ii \,\epsilon } \,+\,\text {(terms regular at $k^2=-m^2$)}\quad . \end{flalign} Note that (8.27) demands two independent conditions, namely that the pole of \(\widetilde {\Delta }^{\mathrm {int}}_F(k) \) is located at \(k^2=-m^2\) and that its residue is \(-\ii \). Inverting \(\widetilde {\Delta }^{\mathrm {int}}_F(k) \), we can disentangle these two conditions into the following two renormalization conditions:

In the next section we will see, by an explicit computation at the \(1\)-loop level, that these two renormalization conditions fix the two counterterms \(\delta _Z\) and \(\delta _m\). In order to link our QFT to the experimentally determined physical coupling \(\lambda \), let us consider the \(2\to 2\) scattering amplitude

\begin{flalign} \braket {q_1,q_2;\mathrm {out}}{k_1,k_2;\mathrm {in}}\,=:\, \mathcal {A}(k_1,k_2,q_1,q_2)\,(2\pi )^4\,\delta (k_1+k_2 - q_1-q_2)\quad , \end{flalign} where, for later convenience, we have factorized off the Dirac delta function that enforces overall relativistic momentum conservation. Motivated by the fact that, at lowest order perturbation theory, the amplitude \(\mathcal {A}(k_1,k_2,q_1,q_2)\) is proportional to the coupling constant via (4.78), it makes sense to formulate a renormalization condition that links the physical coupling \(\lambda \) to \(\mathcal {A}(k_1,k_2,q_1,q_2)\). Since the amplitude (including quantum corrections) will in general depend on the momenta, we have to specify an energy scale at which we link \(\mathcal {A}(k_1,k_2,q_1,q_2)\) to the experimental value of \(\lambda \). The relevant energy scales appearing in \(2\to 2\) scattering are given by the three Mandelstam variables

\begin{flalign} \label {eqn:Mandelstam} s\,:=\,(k_1+k_2)^2~~,\quad t\,:=\, (k_1-q_1)^2~~,\quad u\,:=\,(k_1-q_2)^2\quad , \end{flalign} which (in disguise) already appeared in (4.80), even though we didn’t call them \(s,t,u\) yet. Assuming that our experiment took place at very low energies, e.g. in a normal lab and not in a particle collider, all particles are slow so that their rest energy dominates their spatial momenta. This means that we have approximately \(k_1\approx k_2\approx q_1\approx q_2\approx (m,0,0,0)\), hence \(s=-4m^2\), \(t=0\) and \(u=0\). In this scenario it makes sense to introduce the following renormalization condition

which links the physical coupling constant to the low-energy limit of the scattering amplitude. In the next section we will see, by an explicit computation at the \(1\)-loop level, that this renormalization condition fixes the counterterm \(\delta _\lambda \).

The key feature of renormalizable QFTs is that, once the finitely many counterterms are fixed by finitely many renormalization conditions in terms of finitely many experimentally determined parameters, all time-ordered \(n\)-point functions and scattering amplitudes become finite. In particular, after renormalization has been carried out, the QFT does not have any divergences anymore and it becomes useful for physical predictions. The heuristic explanation for this lies in our analysis of the superficial degree of divergence in Section 8.1, but with some more efforts, which go beyond the scope of this module, one can also prove this rigorously for many examples of QFTs, such as \(\Phi ^4\)-theory, QED and Yang-Mills theory in \(d=4\) spacetime dimensions.