8.3 \(1\)-loop renormalization

In this section we shall make precise in which sense the renormalization conditions allow us to determine the counterterms. For illustrative purposes, we consider again the example of \(\Phi ^4\)-theory in \(d=4\) dimensions, see Section 8.2 for the relevant Feynman rules including counterterms, and focus only on \(1\)-loop corrections. Renormalization at higher orders in perturbation theory is possible but computationally much more involved. If you are interested in these aspects, you can have a look at the book by Peskin/Schroeder (Chapters 10.4 and 10.5), but this material is not relevant for our module.

Fixing the counterterms: Let us focus first on the two renormalization conditions (8.28) and (8.29) for the interacting Feynman propagator. Following similar arguments as in (4.41), we can use perturbation theory (in the form of the Gell-Mann and Low formula and Wick’s theorem) to determine the Feynman diagrams for the interacting propagator and find

Note that the counterterm in the last diagram is a new feature of renormalized perturbation theory that wasn’t previously there in (4.41) where we worked with the bare parameters. The Fourier transform of the interacting Feynman propagator can be determined with a similar calculation as in (4.43) and one finds

\begin{flalign} \widetilde {\Delta }^{\mathrm {int}}_F(k)\,=\, \frac {-\ii }{ k^2 + m^2 + \frac {\lambda }{2}\,\Delta _F(0) +\delta _Z\,k^2 + \delta _m -\ii \,\epsilon }\,+\,\mathcal {O}(\lambda ^2)\quad . \end{flalign} Inverting this expression, we can send \(\epsilon \to 0\) and obtain

\begin{flalign} \widetilde {\Delta }^{\mathrm {int}}_F(k)^{-1} \,=\,\ii \, \bigg ( k^2 + m^2 + \frac {\lambda }{2}\,\Delta _F(0) +\delta _Z\,k^2 + \delta _m \bigg ) \,+\,\mathcal {O}(\lambda ^2)\quad . \end{flalign} The two renormalization conditions (8.28) and (8.29) then fix the two counterterms according to

\begin{flalign} \label {eqn:deltaZdeltam} \delta _Z \,=\,0 + \mathcal {O}(\lambda ^2)~~,\quad \delta _m \,=\,-\frac {\lambda }{2}\,\Delta _F(0) \,+\,\mathcal {O}(\lambda ^2)\quad , \end{flalign} where the loop integral

\begin{flalign} \label {eqn:loopintegral1} \Delta _F(0)\,=\,\lim _{\epsilon \to 0}\, \int _{\bbR ^4}\frac {-\ii }{l^2+m^2-\ii \,\epsilon }\,\frac {\dd l}{(2\pi )^4} \end{flalign} is given explicitly by the formula (3.57) for the Feynman propagator. We postpone the computation of this loop integral to a later point in this section and focus next on the third renormalization condition (8.32). The \(2\to 2\) scattering amplitude can be computed perturbatively using the Feynman rules with counterterms from Section 8.2 (see also Section 4.6 for further details) and one finds

Note that the three loop diagrams are all similar and related to each other by exchanging the external momenta \(k_1,k_2,q_1,q_2\). Looking for instance at the first loop diagram, we can compute via the Feynman rules

Denoting the \(\epsilon \to 0\) limit of this loop integral by

\begin{flalign} \label {eqn:loopintegral2} \ii \,V(p^2)\,:=\,\lim _{\epsilon \to 0}\, \int _{\bbR ^4} \frac {-\ii }{l^2 + m^2-\ii \,\epsilon }\, \frac {-\ii }{(p-l)^2 + m^2-\ii \,\epsilon } \,\frac {\dd l}{(2\pi )^4}\quad , \end{flalign} we obtain for the 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) \end{flalign} the following expression

\begin{flalign} \nn \mathcal {A}(k_1,k_2,q_1,q_2)\,&=\, -\ii \,\lambda + \frac {(-\ii \,\lambda )^2}{2}\,\Big (\ii \,V(s) + \ii \,V(t) + \ii \,V(u)\Big ) - \ii \,\delta _\lambda \,+\,\mathcal {O}(\lambda ^3)\\[3pt] \,&=\, -\ii \, \bigg ( \lambda + \frac {\lambda ^2}{2}\,\Big (V(s) + V(t) + V(u)\Big ) + \delta _\lambda \bigg ) \,+\,\mathcal {O}(\lambda ^3)\quad , \end{flalign} where we recall that \(s,t,u\) denote the three Mandelstam variables (8.31). Our third renormalization condition (8.32) then fixes the counterterm according to

\begin{flalign} \label {eqn:deltalambda} \delta _\lambda \,=\,-\frac {\lambda ^2}{2}\,\Big (V(-4m^2) +2\,V(0)\Big )\,+\,\mathcal {O}(\lambda ^3)\quad . \end{flalign} Summing up, with equations (8.39) and (8.45) we achieved our goal to fix all three counterterms (\(\delta _Z\), \(\delta _m\) and \(\delta _\lambda \)) in terms of loop integrals and the given physical parameters.

Towards computing the loop integrals: It remains to determine the loop integrals in (8.40) and (8.43) that enter our counterterms above. Since these integrals are divergent (at least superficially, as one easily checks by power counting), one should not expect to find a numerical value, but rather one should analyze their divergent behavior. In order to do so, one has to introduce a suitable regulator that makes these integrals finite and characterize which kinds of divergences arise in the limit of removing the regulator. The simplest regulator is a momentum space cut-off, i.e. one restricts the integrations to loop momenta whose norm is less than some constant \(\Lambda >0\), and then investigates the divergences arising in the limit \(\Lambda \to \infty \). (Recall that we have done something like this in our heuristic study of the superficial degree of divergence, see (8.7).) Since momentum space cut-offs are cumbersome to work with, and also not suitable for gauge theories because they break gauge invariance, QFT practitioners have invented smarter regularization techniques, such as dimensional regularization which we shall use below. The basic idea behind dimensional regularization is to consider an analytic extension of the loop integrals to complex spacetime dimensions \(d\in \bbC \), where they become finite, and then study the divergences that arise when sending the dimension \(d\to 4\) or, more generally, to the desired integer dimension one is interested in. This may sound a bit crazy, but as we shall see below this method is very convenient for QFT computations. Please note that dimensional regularization is just a mathematical trick to regularize a QFT! One should not philosophize about what one means by physics in dimension \(d=\sqrt {2} + \ii \,\pi \).

Dimensional regularization is a technique that applies to \(d\)-dimensional integrals of the form

\begin{flalign} \label {eqn:generalintegral} I_{d,n}(\Delta )\,:=\,\int _{\bbR ^d} \frac {1}{(l^2 +\Delta )^n}\,\frac {\dd l}{(2\pi )^d}\quad , \end{flalign} where \(\Delta \) is some complex parameter. While our first loop integral (8.40) is clearly of this type (set \(n=1\) and \(\Delta = m^2 - \ii \,\epsilon \)), it is not immediately obvious that our second loop integral (8.43) is too. There is a nice trick, going back to Feynman, that allows us to rewrite (8.43) as an integral of the form (8.46). This trick is based on the following observation.

Using Lemma 8.9, we can rewrite the \(d\)-dimensional generalization of our second loop integral (8.43) as follows:

\begin{flalign} \nn \ii \,V(p^2) \,&=\,-\lim _{\epsilon \to 0}\,\int _{\bbR ^d}\frac {1}{l^2 + m^2-\ii \,\epsilon } \, \frac {1}{(p-l)^2 + m^2-\ii \,\epsilon }\,\frac {\dd l}{(2\pi )^d}\\[4pt] \nn \,&=\,-\lim _{\epsilon \to 0}\,\int _{\bbR ^d}\int _{0}^1 \frac {1}{\big (l^2 - 2\,(1-\alpha )\,p\,l + (1-\alpha )\,p^2 + m^2-\ii \,\epsilon \big )^2}~\dd \alpha \,\frac {\dd l}{(2\pi )^d}\\[4pt] \nn \,&=\,-\lim _{\epsilon \to 0}\,\int _{\bbR ^d}\int _{0}^1 \frac {1}{\big (k^2 +\alpha \, (1-\alpha )\,p^2 + m^2-\ii \,\epsilon \big )^2}~\dd \alpha \,\frac {\dd k}{(2\pi )^d}\\[4pt] \,&=\,-\lim _{\epsilon \to 0}\,\int _{0}^1 \bigg ( \int _{\bbR ^d}\frac {1}{\big (l^2 +\alpha \, (1-\alpha )\,p^2 + m^2-\ii \,\epsilon \big )^2}~\frac {\dd l}{(2\pi )^d}\bigg ) \,\dd \alpha \quad .\label {eqn:loopintegral2feynmanparameter} \end{flalign} In the third step we substituted \(k=l-(1-\alpha )\,p\) and in the last step we relabeled \(k\) back to \(l\) and also exchanged the order of integrations. (Note that there is no theorem telling us that we are allowed to exchange the order of integrations, but this step is part of the regularization process.) The integral in the parenthesis is of the form (8.46) with \(n=2\) and \(\Delta = \alpha \, (1-\alpha )\,p^2 + m^2-\ii \,\epsilon \).

The master integral (8.46): Since our desired loop integrals (8.40) and (8.50) are special cases of (8.46), it remains to be shown how this general integral can be analyzed. Looking at the integrand, we note that it has poles at

\begin{flalign} l^2 +\Delta = -(l^0)^2 + \mathbf {l}^2 +\Delta \,=\,0 \quad \Longleftrightarrow \quad l^0 = \pm \sqrt { \mathbf {l}^2 +\Delta }\quad . \end{flalign} To understand the location of these poles in the complex plane, we note that in all our examples above the imaginary part of \(\Delta \) is negative, i.e. \(\mathrm {Im}(\Delta )<0\), as a consequence of the \(-\ii \,\epsilon \) terms in the Feynman propagator. Hence, also the imaginary part of \(\mathbf {l}^2 +\Delta \) is negative because \(\mathbf {l}^2\) is real. Writing this complex number in polar form, we get \(\mathbf {l}^2 + \Delta = \vert \mathbf {l}^2 + \Delta \vert ~e^{\ii \,\phi }\) with a phase \(\phi \in (\pi ,2\pi )\). The two square roots of this complex number are then given by

\begin{flalign} \pm \,\sqrt {\mathbf {l}^2 + \Delta }\,=\,\begin{cases} \vert \mathbf {l}^2 + \Delta \vert ^{1/2}~e^{\ii \,\frac {\phi }{2}}\\ \vert \mathbf {l}^2 + \Delta \vert ^{1/2}~e^{\ii \,(\frac {\phi }{2} +\pi )} \end {cases} \end{flalign} with phases \(\frac {\phi }{2}\in (\frac {\pi }{2},\pi )\) and \(\frac {\phi }{2} +\pi \in (\frac {3\pi }{2},2\pi )\), i.e. the poles are located in the second and the fourth quadrant of the complex plane. If we now consider the following contour

then Cauchy’s residue theorem implies that the contour integral

\begin{flalign} \oint _\gamma \frac {1}{\big ((-l^0)^2 + \mathbf {l}^2 + \Delta \big )^n}~\dd l^0\,=\,0 \end{flalign} vanishes because all poles are located in the red regions, hence no pole gets encircled by the contour \(\gamma \). Sending the curved parts of this contour to infinity, their contributions to the integral become zero because the integrand falls off like \(\sim \frac {1}{(l^0)^{2n}}\). Hence, we obtain the identity

\begin{flalign} \int _{-\infty }^\infty \frac {1}{\big ((-l^0)^2 + \mathbf {l}^2 + \Delta \big )^n}~\dd l^0 \,+\, \int _{\ii \infty }^{-\ii \infty }\frac {1}{\big ((-l^0)^2 + \mathbf {l}^2 + \Delta \big )^n}~\dd l^0\,=\,0\quad , \end{flalign} which tells us that the integral over the real \(l^0\) axis (which we would like to compute) can be computed by an integral over the imaginary \(l^0\) axis. Introducing the new variable \(l_E^0 := \ii \,l^0\), we can write this equivalently as

\begin{flalign} \int _{-\infty }^\infty \frac {1}{\big ((-l^0)^2 + \mathbf {l}^2 + \Delta \big )^n}~\dd l^0 \,=\,\ii \,\int _{-\infty }^{\infty }\frac {1}{\big ((l_E^0)^2 + \mathbf {l}^2 + \Delta \big )^n}~\dd l_E^0\quad . \end{flalign} This reformulation of the integral is often called Wick rotation and the vector

\begin{flalign} l_E\,:=\, \begin{pmatrix} l_E^0\\ \mathbf {l} \end {pmatrix}\,\in \,\bbR ^d \end{flalign} is called the Euclidean momentum, which is motivated by the fact that the integrand depends on the Euclidean (in contrast to the Minkowski) norm square

\begin{flalign} l_E^2 \,:=\, (l_E^0)^2 + \mathbf {l}^2 \quad . \end{flalign} Summing up, we have shown that our master integral (8.46) can be computed equivalently in \(d\)-dimensional Euclidean space via

where \(\dd l_E := \dd l_E^0\,\dd l^1\,\cdots \,\dd l^{d-1} \) denotes the \(d\)-dimensional Euclidean volume element.

Since the Wick-rotated integral (8.59) is invariant under \(d\)-dimensional rotations, it makes sense to work with \(d\)-dimensional spherical coordinates. This yields

\begin{flalign} \nn I_{d,n}(\Delta )\,&=\, \ii \,\int _{\bbR ^d} \frac {1}{(l_E^2 +\Delta )^n}\,\frac {\dd l_E}{(2\pi )^d} \,=\,\frac {\ii }{(2\pi )^d}\,\bigg ( \int _{\mathbb {S}^{d-1}}\dd \Omega _{d-1}\bigg )~\int _0^{\infty }\frac {\rho ^{d-1}}{(\rho ^2 + \Delta )^n}\,\dd \rho \\ \,&=\,\frac {\ii }{2\,(2\pi )^d}\,\bigg (\int _{\mathbb {S}^{d-1}}\dd \Omega _{d-1}\bigg )~\int _0^{\infty }\frac {(\rho ^2)^{\frac {d-2}{2}}}{(\rho ^2 + \Delta )^n}\,\dd (\rho ^2)\quad ,\label {eqn:generalintegralspherical} \end{flalign} where \(\rho = \sqrt {l_E^2} = \sqrt {(l_E^0)^2 + \mathbf {l}^2 }\) is the radius coordinate and \(\dd \Omega _{d-1}\) denotes the area element of the \(d{-}1\)-dimensional unit sphere \(\mathbb {S}^{d-1} = \{x\in \bbR ^d \,:\, x^2 =1\}\) .

Using this lemma, we continue our calculation in (8.60) and obtain

\begin{flalign} \nn I_{d,n}(\Delta )\,&=\,\frac {\ii }{2\,(2\pi )^d}\,\bigg (\int _{\mathbb {S}^{d-1}}\dd \Omega _{d-1}\bigg )~\int _0^{\infty }\frac {(\rho ^2)^{\frac {d-2}{2}}}{(\rho ^2 + \Delta )^n}\,\dd (\rho ^2)\\ \nn \,&=\, \frac {\ii }{(4\pi )^{\frac {d}{2}}\,\Gamma (\tfrac {d}{2})}~\int _0^{\infty }\frac {(\rho ^2)^{\frac {d-2}{2}}}{(\rho ^2 + \Delta )^n}\,\dd (\rho ^2)\\ \,&=\,\frac {\ii }{(4\pi )^{\frac {d}{2}}\,\Gamma (\tfrac {d}{2})}~\frac {1}{\Delta ^{n-\frac {d}{2}}}~ \int _0^{1} t^{n-\frac {d}{2}-1}\,(1-t)^{\frac {d}{2}-1}\,\dd t\quad ,\label {eqn:generalintegralspherical2} \end{flalign} where in the last step we changed the integration variables according to \(t=\Delta /(\rho ^2+\Delta )\). Let us observe that the remaining integral in (8.64) is related to the well-studied Euler beta function

\begin{flalign} B(z_1,z_2)\,:=\,\int _0^1 t^{z_1-1}\,(1-t)^{z_2-1}\quad , \end{flalign} which itself is related to the gamma function by the identity

\begin{flalign} B(z_1,z_2)\,=\,\frac {\Gamma (z_1)\,\Gamma (z_2)}{\Gamma (z_1+z_2)}\quad . \end{flalign} This allows us to express (8.64) in the following very useful final form

which involves the well-studied gamma function.

Computing the loop integrals: With these preparations, we are now finally in the position to compute our loop integrals (8.40) and (8.50). For a reason that will become clear in a moment, we will keep the dimension \(d\) as a free parameter and investigate only later what happens when \(d\to 4\) approaches the desired spacetime dimension. For the first loop integral (8.40), we find

\begin{flalign} \nn \Delta _F(0)\,&=\,\lim _{\epsilon \to 0}\, \int _{\bbR ^d}\frac {-\ii }{l^2+m^2-\ii \,\epsilon }\,\frac {\dd l}{(2\pi )^d} \,=\,-\ii \,\lim _{\epsilon \to 0}I_{d,1}\big (m^2-\ii \,\epsilon \big )\\[4pt] \,&=\, \lim _{\epsilon \to 0} \,\frac {1}{(4\pi )^{\frac {d}{2}}} ~\frac {\Gamma (1-\tfrac {d}{2})}{\Gamma (1)}~ \frac {1}{\big (m^2-\ii \,\epsilon \big )^{1-\frac {d}{2}}}\,=\, \frac {m^{d-2}}{(4\pi )^{\frac {d}{2}}} ~\Gamma (1-\tfrac {d}{2})\quad ,\label {eqn:loopintegral1final} \end{flalign} where in the third step we used our general master integral formula (8.66), and in the last step we took the limit \(\epsilon \to 0\) and used \(\Gamma (1) = 0!=1\). (Recall that \(\Gamma (n) = (n-1)!\) for any positive integer \(n\).) Similarly, we find for the second loop integral (8.50)

\begin{flalign} \nn \ii \,V(p^2) \,&=\,-\lim _{\epsilon \to 0}\,\int _{0}^1 \bigg ( \int _{\bbR ^d}\frac {1}{\big (l^2 +\alpha \, (1-\alpha )\,p^2 + m^2-\ii \,\epsilon \big )^2}~\frac {\dd l}{(2\pi )^d}\bigg ) \,\dd \alpha \\[4pt] \nn \,&=\,-\lim _{\epsilon \to 0}\,\int _{0}^1 I_{d,2}\big (\alpha \, (1-\alpha )\,p^2 + m^2-\ii \,\epsilon \big ) ~\dd \alpha \\[4pt] \nn \,&=\, \lim _{\epsilon \to 0}\,\int _{0}^1 \frac {-\ii }{(4\pi )^{\frac {d}{2}}}\,\frac {\Gamma (2-\tfrac {d}{2})}{\Gamma (2)}~\frac {1}{\big (\alpha \, (1-\alpha )\,p^2 + m^2-\ii \,\epsilon \big )^{2-\frac {d}{2}}}~\dd \alpha \\ \,&=\, \int _{0}^1\frac {-\ii }{(4\pi )^{\frac {d}{2}}} \, \frac {\Gamma (2-\tfrac {d}{2})}{\big (\alpha \, (1-\alpha )\,p^2 + m^2\big )^{2-\frac {d}{2}}}~\dd \alpha \quad .\label {eqn:loopintegral2final} \end{flalign} From these expressions we can now finally identify the divergences of the loop diagrams. Recall that the gamma function \(\Gamma (z)\) has poles at every non-positive integer \(z=0,-1,-2,\dots \). If we now try to insert \(d=4\) into (8.67) and (8.68), we obtain divergent expressions due to the poles \(\Gamma (1-\tfrac {d}{2}) = \Gamma (-1)\) and \(\Gamma (2-\frac {d}{2})=\Gamma (0)\). The main trick of dimensional regularization is to consider a dimension \(d=4-\varepsilon \) that is slightly smaller than our desired \(4\) spacetime dimensions, i.e. \(\varepsilon >0\) is very small, and characterize these divergences by studying a Laurent expansion of the relevant gamma functions around \(\varepsilon =0\). (The parameter \(\varepsilon \) is unrelated to the \(-\ii \,\epsilon \) term in the Feynman propagator, which is why it is denoted by a slightly different symbol. Luckily there is little chance of confusion because we have already carried out the limit \(\epsilon \to 0\), so all \(\varepsilon = 4-d\) below are the ones associated with the spacetime dimension.) The Laurent expansions that are relevant for us can be looked up in the QFT literature (see e.g. Peskin/Schroeder (Appendix A.4)) and they read as follows

\begin{flalign} \Gamma (1-\tfrac {d}{2}) \,&=\, \Gamma (-1 + \tfrac {\varepsilon }{2}) \,=\, -\bigg (\frac {2}{\varepsilon } + 1 -\gamma \bigg ) \,+\,\mathcal {O}(\varepsilon ) \quad ,\\ \Gamma (2-\tfrac {d}{2}) \,&=\, \Gamma (\tfrac {\varepsilon }{2}) \,=\, \frac {2}{\varepsilon } - \gamma \,+\,\mathcal {O}(\varepsilon )\quad , \end{flalign} where \(\gamma \approx 0.5772\) is the Euler-Mascheroni constant. Using also a Taylor expansion of the form

\begin{flalign} A^{b\,\varepsilon }\,=\, e^{b\,\varepsilon \,\log (A)} \,=\, 1 + b\,\varepsilon \,\log (A) + \mathcal {O}(\varepsilon ^2) \end{flalign} for \(m^{d-2}\) and \((4\pi )^{-\frac {d}{2}}\) in (8.67), and for \((4\pi )^{-\frac {d}{2}}\) and \(\big (\alpha \, (1-\alpha )\,p^2 + m^2\big )^{\frac {d}{2}-2}\) in (8.68), we obtain the Laurent expansions

and

As a consequence of the \(\frac {1}{\varepsilon }\)-poles, these loop integrals are divergent when sending \(\varepsilon \to 0\), which was expected from our heuristic analysis of their superficial degree of divergence.

Conclusions: Inserting (8.71) and (8.72) into our formulas for the counterterms (8.39) and (8.45), we obtain

As expected, the counterterms are divergent (in the limit \(\varepsilon \to 0\)), which is however not a problem since, by design, they should cancel the divergences appearing in loop diagrams such that the time-ordered \(n\)-point functions and scattering amplitudes are finite. To see an explicit example of such cancellations, let us consider again the \(2\to 2\) scattering amplitude

given explicitly in (8.44). Inserting our results for the loop integral (8.72) and the counterterm (8.73) into this expression, we find

\begin{flalign} \nn \mathcal {A}(k_1,k_2,q_1,q_2)\,&=\,-\ii \,\lambda -\frac {\ii \,\lambda ^2}{32\pi ^2} \,\int _0^1\bigg (\log \bigg (\frac {m^2 + \alpha \,(1-\alpha )\,s}{m^2 - \alpha \,(1-\alpha )\,4m^2}\bigg ) + \log \bigg (\frac {m^2 + \alpha \,(1-\alpha )\,t}{m^2}\bigg )\\[3pt] &\qquad \qquad \qquad +\log \bigg (\frac {m^2 + \alpha \,(1-\alpha )\,u}{m^2}\bigg ) \bigg )~\dd \alpha \,+ \mathcal {O}(\lambda ^3) + \mathcal {O}(\varepsilon )\quad , \end{flalign} where we recall the Mandelstam variables \(s=(k_1+k_2)^2\), \(t=(k_1-q_1)^2\) and \(u=(k_1-q_2)^2\). Note that this scattering amplitude does not contain any \(\tfrac {1}{\varepsilon }\)-poles, which means that we can remove the regulator by sending \(\varepsilon \to 0\). The underlying mechanism is that the \(\tfrac {1}{\varepsilon }\)-poles in the loop diagrams are canceled precisely by the poles in the counterterm (8.73), leading to a finite result. It is important to emphasize that the scattering amplitude is finite for all momenta \(k_1,k_2,q_1,q_2\) and not only at the particular energy scale at which we have introduced the renormalization condition (8.32). With more sophisticated techniques, one can show that such cancellations of divergences happen in all time-ordered \(n\)-point functions and all scattering amplitudes, i.e. the renormalization process successfully cured the ultraviolet divergences of our QFT.

Further reading

For more details about renormalization, see e.g. Peskin/Schroeder (Chapter 10), Nastase (Chapters 32–35) and Srednicki (Chapters 14–18) from our reading list in Section 1.3.