Lecture Notes for MATH4017 Quantum Field Theory

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Chapter 8 Renormalization

The perturbative approach to interacting QFT that we have developed in Chapter 4 is still incomplete and, even worse, seemingly inconsistent as a consequence of potential divergences in loop diagrams that contribute at higher orders in perturbation theory. See in particular Warning 4.5 for an illustration. This chapter provides a very brief introduction to renormalization, which is an important and successful technique that allows one to “cure” ultraviolet divergences by absorbing them into a suitable redefinition of the parameters of the QFT. To simplify our presentation, we consider only scalar QFTs, but it is important to stress that, with some additional computational efforts, other QFTs, e.g. quantum electrodynamics from Chapter 7, can be renormalized as well. Renormalization is not only a mathematical trick that leads to a well-defined perturbation theory, but it also has direct physical consequences, for instance in the form of “running” (i.e. energy-scale dependent) coupling constants. This can be formalized by so-called renormalization group equations, which is an advanced topic that we shall only touch very briefly in Remark 8.8. It is also important to stress that renormalization only deals with ultraviolet (i.e. short distance = large momenta) divergences, while the treatment of infrared (i.e. long distance = small momenta) divergences requires different techniques that will not be discussed in these lecture notes. To avoid the appearance of infrared divergences, we shall always work with a massive scalar field.

8.1 Superficial divergence and power counting

Before discussing the renormalization of ultraviolet divergences in detail, it will be useful to develop a heuristic but useful tool that allows us to identify the divergent Feynman diagrams. To simplify our presentation, we consider only the case of a scalar field theory on the $d$-dimensional Minkowski spacetime $(\bbR ^d,\eta )$ with action

\begin{flalign} \label {eqn:actionbare} S[\Phi ] \,=\,\int _{\bbR ^d} -\bigg (\frac {1}{2}\,\partial ^\mu \Phi \,\partial _\mu \Phi + \frac {m_0^2}{2}\,\Phi ^2 + \frac {\lambda _0}{N!}\,\Phi ^N\bigg )\,\dd x\quad . \end{flalign} In this chapter it will be convenient to decorate the parameters $m_0^2$ and $\lambda _0$ of the classical action with a subscript ${}_0$ in order to distinguish them from the corresponding physical/renormalized parameters, see e.g. Observation 4.7 and Section 8.2 below. (In QFT jargon, the parameter $m_0^2$ is called the bare mass and $\lambda _0$ the bare coupling constant.) Associated with any Feynman diagram of this theory (see Section 4.6 for the Feynman rules) are the following characteristic numbers:

\begin{flalign} E \,&=\,\text {number of external lines}\quad ,\\ P \,&=\,\text {number of internal lines/propagators}\quad ,\\ V \,&=\, \text {number of vertices}\quad ,\\ L \,&=\, \text {number of loops}\quad . \end{flalign} From the Feynman rules, we see that each internal line comes with an undetermined momentum $l\in \bbR ^d$, together with a momentum space integral $\int _{\bbR ^d} \frac {\dd l}{(2\pi )^d}$, and each vertex comes with a Dirac delta function $\delta (p_{\mathrm {in}}-p_{\mathrm {out}})$ that enforces relativistic momentum conservation at this vertex. The $V$ many delta functions kill $V-1$ of the $P$ many momentum space integrations, leaving one Dirac delta function that enforces overall relativistic momentum conservation. Hence, we obtain the identity

\begin{flalign} \label {eqn:superficialtmp1} L \,=\, P-V+1 \end{flalign} that expresses the number of loop integrals $L$ in terms of $P$ and $V$. Furthermore, since each external line ends with $1$ end at a vertex and each internal line ends with $2$ ends, we have

\begin{flalign} \label {eqn:superficialtmp2} N\,V = 2\,P + E\quad , \end{flalign} where the $N$ is due to the fact that $\Phi ^N$-theory has an $N$-valent interaction vertex. For large loop momenta $l_1,\dots ,l_L\in \bbR ^d$, the structure of the momentum space integral associated with our Feynman diagram looks schematically like

\begin{flalign} \int _{\bbR ^{L d}} \frac {1}{l_1^2\cdots l_P^2}\,\dd l_1\cdots \dd l_L\quad , \end{flalign} i.e. each loop momentum contributes a $d$-dimensional volume element $\dd l$ and each internal line contributes a $1/l^2$. (Note that, for large loop momenta, the masses and external momenta in Feynman propagators can be neglected, e.g. $1/(l^2 + m_0^2)\approx 1/l^2$.) To obtain a quantity that allows for a rough estimate of the divergence/convergence property of such integrals, we compare the power of the momenta in the numerator and the denominator.

Definition 8.1. The superficial degree of divergence of a Feynman diagram is defined as
$\seteqnumber{0}{8.}{5}$
\begin{flalign} \label {eqn:superficial} D\,:=\, d\,L - 2\,P\quad . \end{flalign}

Continuing with our heuristic analysis, we introduce a momentum space cut-off, i.e. we integrate only over momenta whose norm is less than some constant $\Lambda >0$, and thereby find the following behavior

\begin{flalign} \int _{\bbR ^{L d}}^{\Lambda } \frac {1}{l_1^2\cdots l_P^2}\,\dd l_1\cdots \dd l_L \,\sim \,\begin{cases} \Lambda ^D &~,~~\text {for }D>0\quad ,\\ \log (\Lambda )&~,~~\text {for }D=0\quad ,\\ \Lambda ^{-\vert D\vert }&~,~~\text {for }D<0\quad .\label {eqn:superficialdivergenceregulator} \end {cases} \end{flalign} This heuristic power counting analysis can be made more precise by using the concept of Wick rotation (see around Eqn. (8.59) below) to turn the integral (8.7) into an Euclidean momentum space integral. This extra level of precision is however not necessary for understanding the main message of the present section. According to this heuristic reasoning, one finds that Feynman diagrams with $D\geq 0$ are divergent when sending the cut-off $\Lambda \to \infty $, while those with $D<0$ are finite. Unfortunately, the superficial degree of divergence $D$ does not always reflect the actual divergence or finiteness property of a Feynman diagram, scattering amplitude or time-ordered $n$-point function. Indeed, the following exceptions might happen:

1. In the superficially divergent case $D\geq 0$, infinite terms might cancel each other and lead to a finite end result. This typically happens in gauge theories or in QFTs with supersymmetry.
2. In the superficially finite case $D<0$, a diagram might be divergent if the divergence is due to a simpler divergent subdiagram. A simple example for this phenomenon is given by the Feynman diagram

$(8.8) \{begin}{flalign} \label {eqn:subdivergence} \parbox {4cm}{ \begin{tikzpicture} \begin{feynman} \vertex (i1) at (-1.5,1); \vertex (i2) at (-1.5,-1); \vertex (f1) at (1.5,1.5); \vertex (f2) at (1.5,0.5); \vertex (f3) at (1.5,-0.5); \vertex (f4) at (1.5,-1.5); \vertex (a) at (0,0.75); \vertex (b) at (0,-0.75); \vertex (c) at (0,0); \vertex (d) at (-0.5,0); \diagram *{ (i1) -- (a), (a) -- (f1), (a) -- (f2), (i2) -- (b), (b) -- (f3), (b) -- (f4), (a) -- (b), (c) -- [out=135, in=90] (d), (c) -- [out=225, in=-90] (d), }; \draw [fill=black] (a) circle(0.75mm); \draw [fill=black] (b) circle(0.75mm); \draw [fill=black] (c) circle(0.75mm); \end {feynman} \end {tikzpicture} } \{end}{flalign}$

in $\Phi ^4$-theory in $d=4$ dimensions. This diagram has $L=1$ loops and $P=3$ internal lines, hence $D = 4\times 1 - 2\times 3 = -2$. However, as we shall see in Section 8.3, the tadpole loop on the internal propagator is divergent, hence the overall diagram is divergent too.

The first exception is pretty harmless, because it simply means that the superficial degree of divergence might overestimate the number of divergent diagrams. The second exception is potentially quite serious, because it means that, by looking only at the superficial degree of divergence, we might miss some of the divergent diagrams. While a general treatment of subdivergences of Feynman diagrams is possible, it is unfortunately rather technical and hence beyond the scope of this module. In our simple example (8.8), the subdivergence of this diagram will simply be removed by renormalizing the interacting Feynman propagator.

Leveraging the identities (8.3) and (8.4), we can express the superficial degree of divergence (8.6) in the following more useful form

\begin{equation} \label {eqn:superficialuseful} D \,=\,d + \Big (N\,\tfrac {d-2}{2} -d \Big )\,V - \tfrac {d-2}{2}\,E \end{equation}

in terms of the number of vertices $V$ and the number of external lines $E$. From this expression one sees that the prefactor $N\,\tfrac {d-2}{2} -d$ of $V$ is crucial: If this prefactor is negative, then Feynman diagrams with sufficiently many vertices become superficially finite (i.e. $D<0$), hence the QFT has only a finite number of superficially divergent diagrams. On the other hand, if this prefactor is positive, then all Feynman diagrams with a sufficiently high number of vertices $V$ will become superficially divergent (i.e. $D\geq 0$). If the prefactor of $V$ is $0$ (which can only happen for $d\geq 3$), then $D$ does not depend on the number of vertices but only on the number $E$ of external lines. Since the prefactor of $E$ is negative in this case, we have that $D<0$ for sufficiently large $E$. This means that only finitely many scattering amplitudes and/or time-ordered $n$-point functions, namely those with a sufficiently small number $E$ of external lines, are superficially divergent. It is important to distinguish between these three different possibilities, which is done by introducing the following terminology.

Definition 8.2. A QFT is called
- • (power counting) super-renormalizable if only a finite number of connected Feynman diagrams superficially diverge,
- • (power counting) renormalizable if only a finite number of time-ordered $n$-point functions and/or scattering amplitudes superficially diverge (note that these can consist of infinitely many superficially divergent connected Feynman diagrams),
- • (power counting) non-renormalizable if infinitely many time-ordered $n$-point functions and/or scattering amplitudes are superficially divergent at a sufficiently high order in perturbation theory.
Note that
$\seteqnumber{0}{8.}{9}$
\begin{flalign} \text {super-renormalizable}~~\Longrightarrow ~~\text {renormalizable}\quad , \end{flalign} hence super-renormalizable QFTs are a (very special) subclass of the renormalizable ones.

Remark 8.3. There is a huge qualitative difference between renormalizable and non-renormalizable QFTs. For the former type there are only finitely many divergent time-ordered $n$-point functions and/or scattering amplitudes, while for the latter there are infinitely many. Using the renormalization techniques to be developed in this chapter, we will be able to “absorb” these divergences into a redefinition of the parameters of the QFT. For renormalizable QFTs, this will result in a finite number of free parameters, which have to be fixed from experimental input. On the other hand, for non-renormalizable QFTs, absorbing the infinitely many divergences will result in an infinite number of free parameters, which makes such theories non-predictive as all these parameters must be fixed by experiment. As a consequence, physicists are typically looking for renormalizable QFTs to describe nature, but also non-renormalizable QFTs appear in certain areas of low-energy physics as “effective QFTs”.

Warning 8.4. As already mentioned above, the superficial degree of divergence does not necessarily reflect the actual divergence or finiteness properties of a QFT. This means that Definition 8.2 of (power counting) super-renormalizable, renormalizable and non-renormalizable QFTs should be used with some care. In my opinion, this definition is best used as a guiding principle to propose and design suitable QFT models which have a chance to be renormalizable in a mathematically strict sense. For many physically relevant QFTs, such as QED and Yang-Mills theory in $d=4$ dimensions, there exist rigorous proofs confirming their renormalizability.

Example 8.5 (Spacetime dimension $d=4$). Let us consider the physically relevant case of $d=4$ spacetime dimensions. Then the superficial degree of divergence in (8.9) specializes to
$\seteqnumber{0}{8.}{10}$
\begin{flalign} D \,=\, 4+ (N-4)\, V -E\quad . \end{flalign} From this we find that $\Phi ^3$-theory (i.e. $N=3$) in $d=4$ dimensions is super-renormalizable and that $\Phi ^4$-theory (i.e. $N=4$) in $d=4$ dimensions is renormalizable. In contrast to this, $\Phi ^N$-theory in $d=4$ dimensions is non-renormalizable for all $N\geq 5$.

Let us discuss the case of $\Phi ^4$-theory in $d =4$ dimensions in more detail. Since the action functional
$\seteqnumber{0}{8.}{11}$
\begin{flalign} \label {eqn:actionbarePhi4} S[\Phi ] \,=\,\int _{\bbR ^4} -\bigg (\frac {1}{2}\,\partial ^\mu \Phi \,\partial _\mu \Phi + \frac {m_0^2}{2}\,\Phi ^2 + \frac {\lambda _0}{4!}\,\Phi ^4\bigg )\,\dd x \end{flalign} is invariant under the sign flip transformation $\Phi \mapsto -\Phi $, all odd time-ordered $n$-point functions must vanish. Using the superficial degree of divergence $D = 4-E$ for this theory, we further see that all time-ordered $n$-point functions with $n=E\geq 5$ are superficially finite. So the only superficially divergent time-ordered $n$-point functions are the $0$-, $2$- and $4$-point functions

$(8.13) \{begin}{flalign} \parbox {8cm}{ \begin{tikzpicture}[scale=0.75] \draw [fill=lightgray] (-4,0) circle (0.5cm); \draw [thick] (-1.5,0) -- (1.5,0); \draw [fill=lightgray] (0,0) circle (0.5cm); \draw [thick] (3,1) -- (4,0); \draw [thick] (3,-1) -- (4,0); \draw [thick] (4,0) -- (5,1); \draw [thick] (4,0) -- (5,-1); \draw [fill=lightgray] (4,0) circle (0.5cm); \draw (-4,-1.5) node {$D=4$}; \draw (0,-1.5) node {$D=2$}; \draw (4,-1.5) node {$D=0$}; \end {tikzpicture} } \{end}{flalign}$

The divergences in the time-ordered $0$-point function are given by vacuum bubbles, hence they describe an unobservable vacuum energy shift which can be ignored. The divergences in the $2$-point function will lead to a renormalization of the mass parameter (see Observation 4.7) and, at higher loop orders, also a wave function renormalization. Finally, the divergences in the $4$-point function will lead to a renormalization of the coupling constant. One of the goals of this chapter is to make these claims precise.

Example 8.6 (Spacetime dimension $d=6$). The superficial degree of divergence (8.9) specializes in $d=6$ dimensions to
$\seteqnumber{0}{8.}{13}$
\begin{flalign} D \,=\, 6 + \big (2\,N -6\big )\,V - 2\,E\quad . \end{flalign} From this we find that $\Phi ^3$-theory (i.e. $N=3$) in $d=6$ dimensions is renormalizable and that $\Phi ^N$-theory in $d=6$ dimensions is non-renormalizable for all $N\geq 4$. Comparing with the previous example in $d=4$, we note that it becomes harder to find renormalizable QFTs when increasing the dimension of spacetime.

Let us discuss the case of $\Phi ^3$-theory in $d=6$ dimensions in more detail. Using the superficial degree of divergence $D = 6-2\,E$ for this theory, we see that all time-ordered $n$-point functions with $n=E\geq 4$ are superficially finite. So the only superficially divergent time-ordered $n$-point functions are the $0$-, $1$-, $2$- and $3$-point functions

$(8.15) \{begin}{flalign} \parbox {11cm}{ \begin{tikzpicture}[scale=0.75] \draw [fill=lightgray] (-8,0) circle (0.5cm); \draw [thick] (-5.5,0) -- (-4,0); \draw [fill=lightgray] (-4,0) circle (0.5cm); \draw [thick] (-1.5,0) -- (1.5,0); \draw [fill=lightgray] (0,0) circle (0.5cm); \draw [thick] (2.5,0) -- (4,0); \draw [thick] (4,0) -- (5,1); \draw [thick] (4,0) -- (5,-1); \draw [fill=lightgray] (4,0) circle (0.5cm); \draw (-8,-1.5) node {$D=6$}; \draw (-4,-1.5) node {$D=4$}; \draw (0,-1.5) node {$D=2$}; \draw (4,-1.5) node {$D=0$}; \end {tikzpicture} } \{end}{flalign}$

Remark 8.7. The relevant prefactor $N\,\tfrac {d-2}{2} -d$ of $V$ in (8.9) is related to the physical dimension of the coupling constant $\lambda _0$ in the action (8.1). Since we work in natural units $c=1$ and $\hbar =1$, we can measure all quantities (length, time, momentum, energy, …) in the dimension of mass. For example, the mass dimension of the spacetime coordinates is $[x^\mu ]=-1$, which implies that the $d$-dimensional volume element has mass dimension $[\dd x]=-d$, and the one of the derivative is $[\partial _\mu ] =1$. Since the action must be dimensionless, we find that the scalar field has mass dimension
$\seteqnumber{0}{8.}{15}$
\begin{flalign} [\Phi ] = \frac {d-2}{2}\quad . \end{flalign} For the parameters in the action, we then find $[m_0^2]=2$, as expected, and
$\seteqnumber{0}{8.}{16}$
\begin{flalign} [\lambda _0] = d-N\,\frac {d-2}{2}\quad , \end{flalign} which is the additive inverse of the prefactor $N\,\tfrac {d-2}{2} -d$ of $V$ in (8.9). Summing up, this means that super-renormalizable theories have coupling constants with positive mass dimension $[\lambda _0]>0$, renormalizable theories have dimensionless coupling constants $[\lambda _0]=0$ and non-renormalizable theories have coupling constants with negative mass dimension $[\lambda _0]<0$. This observation is useful in practice when designing examples of action functionals for physical theories, because via this simple dimensional analysis one can get a good hint whether the resulting QFT will be renormalizable or not.