Quantum field theory 1, lecture 28

One-loop corrections in quantum electrodynamics

In this section we consider systematically one loop corrections in perturbative quantum electrodynamics. We have calculated previously a one-loop expression for a process that has no tree-level contribution and found a finite result. In contrast to this, if one finds one-loop corrections to processes that actually do have a tree-level contribution, one finds formally infinite results. To deal with these we need to first carefully introduce a regularization of the theory (we can use dimensional regularization), and in a second step we have to reinterpret what we are doing from a different persepctive. This leads to renormalisation. Specifically, it turns out that the terms appearing in the Lagrangian are in fact themselves subject to corrections from quantum fluctuations from loop diagrams.

Corrected propagators and vertices

Consider again the Lagrangian of QED as written before \[\mathscr{L} = - \bar \Psi \gamma^\mu (\partial_\mu - i e A_\mu) \Psi - m \bar \Psi \Psi - \frac{1}{4} F_{\mu\nu} F^{\mu\nu}.\] This Lagrangian enters the functional integral but it is in fact not directly observable. Corrections from quantum fluctuations are always present and one can in fact only observe asymptotic particles described by the full propagators with fluctuation corrections taken into account, and similarly vertices with corrections taken into account. Specifically, at one loop order we have for the effective photon propagator the tree-level term plus a one-loop term,

and only the sum (including in fact further loop corrections at higher order) can actually describe a propagating photon.

Similarly, the fermion propagator at on-lopp order contains the tree-level term plus a one loop term,

and more terms at higher order. Finally, the vertex is a combination of the tree-level expression and a loop expression,

and more terms would contribute at higher orders.

Modified Lagrangian

Motivated by these considerations we modify the Lagrangian by introducing wave function renormalization factors \(Z_\Psi\) and \(Z_A\) for the fields through the replacements \[\Psi \to \sqrt{Z_\Psi} \Psi, \quad\quad \bar\Psi \to \sqrt{Z_\Psi} \bar\Psi, \quad\quad A_\mu \to \sqrt{Z_A} A_\mu,\] and we replace the fermion mass by \[m \to m + \Delta m,\] and the electric charge by \[e\to e+\Delta e.\] The Lagrangian becomes thus \[\mathscr{L} = - Z_\Psi \bar \Psi \gamma^\mu (\partial_\mu - i (e+\Delta e) \sqrt{Z_A} A_\mu) \Psi - (m+\Delta m) Z_\Psi \bar \Psi \Psi - \frac{1}{4} Z_A F_{\mu\nu} F^{\mu\nu}.\] This is supposed to enter the functional integral of the quantum field theory which has been regularized, for example by putting it on a spacetime lattice, by introducing a UV cutoff, or through dimensional regularization.

The idea is now to determine the parameters \(Z_\Psi\), \(Z_A\) and \(\Delta m\) through physics conditions on the full theory (or some approximation at a given order of perturbation theory to it). For example, \(Z_A\) will be chose such that the corrected photon propagator has the standard residue at the points with \[p^2 = -(p^0)^2 + \mathbf{p}^2 =0.\] Similarly, \(Z_\Psi\) and \(\Delta m\) can be determined such that the fermion propagator including quantum corrections has poles at the frequencies \(p^0\) such that \[p^2 + m^2 = -(p^0)^2 + \mathbf{p}^2 + m^2 =0,\] and that these poles have the standard residue. This is in fact a bit intricate because of infrared divergences and the fact that a charged particle can experimentally not distinguished from a charged particle plus a very soft photon. (These two issues are closely connected and need to be dealt with simultaneously.)

Finally, the correction to the charge \(\Delta e\) will be determined such that the corrected vertex function represents the physical charge \(e\).

Perturbative expansion

So far we have worked at leading order or tree-level and saw there how the QED Lagrangian lead to consistent physics results when loop diagrams could be neglected. In the spirit of perturbation theory we therefore expect that to the leading order in the coupling constant, or in the fine structure constant \[\alpha = \frac{e^2}{4\pi}\] we have simply \(Z_\Psi = Z_A = 1\) and \(\Delta m = \Delta e = 0\). In fact, from inspection of the diagrams we infer \[Z_A -1 = \mathcal{O}(\alpha), \quad\quad\quad Z_\Psi-1 = \mathcal{O}(\alpha), \quad\quad\quad \Delta m = \mathcal{O}(\alpha), \quad\quad \Delta e/e = \mathcal{O}(\alpha).\] The idea is now to allow so-called counterterms for these additional contributions to the QED Lagrangian and to determine them through the physics conditions mentioned above. This must be done consistently in perturbation theory, i. e. when one wants to calculate including the \(\mathcal{O}(\alpha)\) corrections one must take counterterms at that order into account.

Photon self energy

We start with the effective photon propagator. Quantum corrections as the one-loop fermion diagram introduced above can be taken into account through a geometric series by writing the corrected photon propagator in momentum space as (known as Dyson equation) \[\begin{split} G_{\mu\nu}(p) = & \Delta_{\mu\nu}(p) + \Delta_{\mu\rho}(p) \Pi^{\rho\sigma}(p) \Delta_{\sigma\nu}(p) + \ldots \end{split}\] Here we use the free photon propagator (in Lorenz or Landau gauge) \[\Delta_{\mu\nu}(p) = \frac{\mathscr{P}_{\mu\nu}(p)}{p^2-i\epsilon} = \frac{\eta_{\mu\nu}-p_\mu p_\nu/p^2}{p^2-i\epsilon},\] and \(\Pi^{\rho\sigma}(p)\) is known as the photon self-energy. The Dyson equation can be better understood in terms of the inverse corrected propagator obtained by suming the geometric series, \[p^2 \eta^{\mu\nu} - p^\mu p^\nu - \Pi^{\mu\nu}(p)= p^2 \mathscr{P}^{\mu\nu}(p) - \Pi^{\mu\nu}(p)\] In other words, \(- \Pi^{\mu\nu}(p)\) is the quantum correction to the inverse photon propagator! The first term is the microscopic or bare inverse propagator with which we have started our discussion of QED. For the corrected inverse propagator to be gauge invariant, we expect the structure \[\Pi^{\mu\nu}(p) = \Pi(p^2) \left( p^2 \eta^{\mu\nu} - p^\mu p^\nu \right),\] where \(\Pi(p^2)\) is now a scalar function of momentum. In perturbation theory at one-loop order, the photon self-energy, and therefore \(\Pi(p)\) has contributions from the fermion loop diagram, as well as from the counter term \(Z_A-1\). The latter is in fact easy to determine and \[\Pi(p^2) = \Pi_\text{1-loop}(p^2) - (Z_A-1).\] It remains to determine the loop contribution. Using the Feynman rules and dropping terms of higher order in \(\alpha\), we find \[\Pi_\text{1-loop}^{\mu\nu}(p) = (-1) i e^2 \int \frac{d^4 l}{(2\pi)^4} \frac{\text{Tr}\left\{ [-i (\slashed{l}+\slashed{p}) +m] \gamma^\mu [-i \slashed{l} +m] \gamma^\nu \right\}}{[(p+l)^2+m^2-i\epsilon][l^2+m^2-i\epsilon]}.\] The minus sign comes from the closed fermion loop.

Evaluation of loop integral

To perform the Dirac trace one decomposes it into one term with four gamma matrices, \[\text{Tr} \left\{ -(\slashed{l}+\slashed{p}) \gamma^\mu \slashed{l} \gamma^\nu \right\} = 4 \left[ -(l+p)^\nu l^\mu + (l+p) \cdot l \, \eta^{\mu\nu} - (l+p)^\mu l^\nu \right],\] and a term with two gamma matrices, \[\text{Tr} \left\{ m^2 \gamma^\mu \gamma^\nu \right\} = 4 m^2 \eta^{\mu\nu}.\] For the denominator we introduce an integral over a Feynman propagator, \[\frac{1}{[(p+l)^2+m^2-i\epsilon][l^2+m^2-i\epsilon]} = \int_0^1 du \frac{1}{[l^2 + 2 u l \cdot p + u p^2+m^2-i\epsilon]^2}\] Now we can shift the integration variable setting \(k=l+u p\). We obtain, dropping linear terms in \(k\) and going to \(d\) dimensions, \[\Pi_\text{1-loop}^{\mu\nu}(p) = - i 4 e^2 \int \frac{d^d k}{(2\pi)^d} \int_0^1 d u \frac{k^2 \eta^{\mu\nu} - 2 k^\mu k^\nu - u(1-u) p^2 \eta^{\mu\nu} + 2 u(1-u) p^\mu p^\nu + m^2}{\left[ k^2 + u(1-u) p^2 + m^2 - i \epsilon \right]^2}.\] In the next step we replace \(2k^\mu k^\nu \to (2/d) k^2 \eta^{\mu\nu}\) and use the following identity valid in dimensional regularization (exercise) \[\int \frac{d^d k}{(2\pi)^d} \frac{\left( \frac{2}{d} -1 \right)k^2-A}{[k^2+A]^2} = 0.\] This yields \[\Pi_\text{1-loop}^{\mu\nu}(p) = \left( p^2 \eta^{\mu\nu} - p^\mu p^\nu \right) i e^2 \int_0^1 du \int \frac{d^d k}{(2\pi)^d} \frac{8 u(1-u)}{[k^2 + u (1-u) p^2 + m^2 - i \epsilon]^2}.\] Note that this has indeed the tensor structure of the classical inverse propagator for photons, as expected.

Scalar integral

We continue to evaluate the integral over \(k\). We do the Wick rotation to Euclidean frequency, \(k^0=ik^0_\text{E}\) which gives a factor \(i\), and yields thus \[\Pi_\text{1-loop}(p) = - 8 e^2 \int_0^1 du \, u(1-u) \int \frac{d^d k_\text{E}}{(2\pi)^d} \frac{1}{[k_\text{E}^2 + u (1-u) p^2 + m^2 ]^2}.\] For the remaining integral we can now use the formula in \(\eqref{eq:ScalarIntegralsDimReg}\). This gives \[\int \frac{d^d k_\text{E}}{(2\pi)^d} \frac{1}{[k_\text{E}^2+A]^2} = \frac{\Gamma(2-d/2)}{(4\pi)^{d/2}} A^{-(2-d/2)}.\] We need to evaluate this in the vicinity of \(d=4\) and set \(d=4-\varepsilon\). We use \[\Gamma(2-d/2) = \Gamma(\varepsilon/2) = \frac{2}{\varepsilon} - \gamma + \mathcal{O}(\varepsilon),\] and \[(4\pi)^{d/2} = (4\pi)^{2-\varepsilon/2} = 16 \pi^2 e^{-\frac{\varepsilon}{2}\ln(4\pi)} = 16 \pi^2 \left[ 1 - \frac{\varepsilon}{2} \ln(4\pi) + \ldots \right],\] as well as \[A^{-(2-d/2)} = A^{-\varepsilon/2} = 1-\frac{\varepsilon}{2} \ln(A).\] Finally we need to take into account that electric charge is not dimensionless away from \(d=4\). We replace therefore \(e^2\) with \[e^2 \mu^{\varepsilon} = e^2 \left[ 1 + \frac{\varepsilon}{2} \ln(\mu^2) + \ldots \right],\] where \(\mu\) is some parameter with dimension of mass and \(e\) remains dimensionless.

Fixing the counterterm

Combining everything and expanding to leading order in \(\varepsilon\) gives \[\Pi(p) = - \frac{e^2}{\pi^2} \int_0^1 du \, u(1-u) \left[ \frac{1}{\varepsilon} - \frac{1}{2} \ln\left( \frac{u(1-u)p^2+m^2}{4\pi e^{-\gamma} \mu^2} \right) \right] - (Z_A-1).\] One now fixes the counterterm \(Z_A-1\) such that \(\Pi(0)=0\), such that for on-shell photons the propagator is not modified. This leads to \[Z_A-1 = - \frac{e^2}{\pi^2} \int_0^1 du \, u(1-u) \left[ \frac{1}{\varepsilon} - \frac{1}{2} \ln\left( \frac{m^2}{4\pi e^{-\gamma} \mu^2} \right) \right].\] The integral over \(u\) can easily be performed here and gives a finite result. The important thing is that the counterterm is formally infinite due to the UV divergence of the involved momentum integrals.

Result for photon self-energy at one loop

Once the counterterm is subtracted from the self energy, we are actually left with a finite photon self-energy \[\Pi(p) = \frac{e^2}{2\pi^2} \int_0^1 du \, u(1-u) \ln\left( \frac{u(1-u)p^2+m^2}{m^2} \right).\] Indeed this satisfies \(\Pi(0)\) and is also independent of the arbitrary scale \(\mu\) we had to intrduce in an intermediate step. For negative \(p^2=-(p^0)^2 +\mathbf{p}^2\) one finds that \(\Pi(p)\) has a brach cut starting from the point where \[(p^0)^2-\mathbf{p}^2 > 4 m^2.\] This corresponds to the physical process where a virtual photon dissociates into an electron-positron pair. Energetically this is only possible when the virtual photon has an energy of at least twice the electron mass.

Final remark

One can now go on and analyse other loop diagrams in this spirit. The story is in principle similar to what we have seen for the photon self-energy. Counterterms in the form of the terms in the Lagrangian are needed to counterbalace UV divergences, and they can be fixed through physical renormalization conditions. Within perturbation theory one obtains then finite results, except when additional infrared divergences appear. These need a somewhat differnt treatment and have a different physical significance. After proper renormalization one can calculate finite terms, such as the electron magnetic moment. We leave a more detailed discussion of renormalization for a continuation of this lecture course in the next term.