Lecture Notes for MATH4017 Quantum Field Theory — Feynman rules and scattering amplitudes in QED

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7.2 Feynman rules and scattering amplitudes in QED

Using perturbative quantization techniques similar to those that we have developed for scalar fields in Chapter 4, one can determine the interacting time-ordered $n$-point functions and scattering amplitudes in quantum electrodynamics. We will skip redoing many of the tedious details and move quickly to a simpler and more intuitive description in terms of Feynman diagrams.

In order to have a well-defined Feynman propagator for the photon field, we have to take into account the gauge symmetries (7.7) and consider as in Chapter 6 a gauge fixed version of the action functional (7.9). Working in Lorenz gauge, this action is given by

\begin{flalign} \nn \widetilde {S}_{\mathrm {QED}}^{}[\Psi ,\overline {\Psi },A] \,&=\, \int _{\bbR ^d} -\bigg (\overline {\Psi }\,\big (\slashed {D} + m \big )\,\Psi +\frac {1}{4}\, F^{\mu \nu }\,F_{\mu \nu } +\frac {1}{2}\,(\partial _\mu A^\mu )^2\bigg )\,\dd x \\[5pt] \,&=\,\int _{\bbR ^d} -\bigg (\overline {\Psi }\,\big (\slashed {\partial } + m \big )\,\Psi + \frac {1}{2}\,\partial ^\mu A^\nu \,\partial _\mu A_\nu \bigg )\,\dd x ~-~ \ii \,q\,\int _{\bbR ^d} \overline {\Psi }\slashed {A}\Psi \,\dd x\quad ,\label {eqn:QEDactionGF} \end{flalign} together with the constraint $\partial _\mu A^\mu =0$. Note that the first term describes the free Dirac and gauge fixed Maxwell actions, and the second term describes a cubic interaction.

Time-ordered $n$-point functions:

Let us recall from Section 5.4 that the free Dirac Feynman propagator is given by

$(7.15) \{begin}{flalign} S_F(x-y) ~=~ \lim _{\epsilon \to 0} \int _{\bbR ^d} \frac {-\ii \,\big ({-}\ii \,\slashed {k} +m\big )}{k^2 + m^2 -\ii \,\epsilon }~e^{\ii \,k\,(x-y)}\,\frac {\dd k}{(2\pi )^d} ~= \parbox {2.5cm}{ \begin{tikzpicture}[scale=1] \draw [thick,-latex] (0.65,0) -- (-0.1,0); \draw [thick] (0,0) -- (-0.65,0); \draw (0.75,0) node{\footnotesize {$~~y$}}; \draw (-0.75,0) node{\footnotesize {$x~~$}}; \end {tikzpicture}}\quad , \{end}{flalign}$

where the arrow points along the flow of particles (or alternatively against the flow of antiparticles). The free photon Feynman propagator was derived in Section 6.3 and it is given by

$(7.16) \{begin}{flalign} (D_F)_{\mu \nu }(x-y) ~=~ \lim _{\epsilon \to 0} \int _{\bbR ^d} \frac {-\ii \,\eta _{\mu \nu }}{k^2 -\ii \,\epsilon }~e^{\ii \,k\,(x-y)}\,\frac {\dd k}{(2\pi )^d} ~=\parbox {2.5cm}{ \begin{tikzpicture}[scale=1] \tikzset {decoration={snake,amplitude=.8mm,segment length=2.5mm, post length=0mm,pre length=0mm}} \draw [thick,decorate] (0.65,0) -- (-0.65,0); \draw (0.75,0) node{\footnotesize {$~~y$}}; \draw (-0.75,0) node{\footnotesize {$x~~$}}; \end {tikzpicture}}\quad . \{end}{flalign}$

The interaction vertices are generated through the exponential $e^{\ii \,S_{\mathrm {int}}^{}}$ in the Gell-Mann and Low formula (4.16). Since the interaction term $S_{\mathrm {int}}^{} = - \ii \,q\,\int _{\bbR ^d} \overline {\Psi }\slashed {A}\Psi \,\dd x$ in QED is cubic, this yields a $3$-valent vertex of the form

$(7.17) \{begin}{flalign} \parbox {1.75cm}{ \begin{tikzpicture}[scale=1] \tikzset {decoration={snake,amplitude=.8mm,segment length=2.5mm, post length=0mm,pre length=0mm}} \draw [thick,-latex] (-0.6,0.6) -- (-0.25,0.25); \draw [thick] (-0.4,0.4) -- (0,0); \draw [thick,-latex] (0,0) -- (-0.4,-0.4); \draw [thick] (-0.3,-0.3) -- (-0.6,-0.6); \fill (0,0) circle[radius=2pt]; \draw (0,0.25) node{\footnotesize {$~z$}}; \draw [thick,decorate] (0,0) -- (1,0); \end {tikzpicture}} \,=\, q\, \gamma ^\mu \,\int _{\bbR ^d}\dd z\quad . \{end}{flalign}$

Based on these building blocks, we can now draw and evaluate Feynman diagrams for time-ordered $n$-point functions in QED. Let us illustrate this by studying an example: The interacting Dirac Feynman propagator to order $q^3$ in the coupling constant $q$. The relevant Feynman diagrams are

$(7.18a) \{begin}{flalign} \expect {\Omega }{\TO \big (\Psi (x)\,\overline {\Psi }(y)\big )}{\Omega } \, =\, \parbox {1.5cm}{ \begin{tikzpicture}[scale=1.5] \draw [thick,-latex] (0.5,0) -- (-0.05,0); \draw [thick] (0,0) -- (-0.5,0); \end {tikzpicture}} ~+~ \parbox {1.5cm}{ \begin{tikzpicture}[scale=1.5] \tikzset {decoration={snake,amplitude=.6mm,segment length=1.5mm, post length=0mm,pre length=0mm}} \draw [thick,-latex] (0.5,0) -- (-0.1,0); \draw [thick] (0,0) -- (-0.5,0); \draw [thick,-latex] (0.5,0) -- (0.3,0); \draw [thick,-latex] (0,0) -- (-0.45,0); \fill (-0.2,0) circle[radius=1.2pt]; \fill (0.2,0) circle[radius=1.2pt]; \draw [thick,decorate] (-0.2,0) to[out=90,in=90] (0.2,0); \draw (0,-0.12) node{\footnotesize {$~$}}; \end {tikzpicture}} ~+~ \parbox {1.5cm}{ \begin{tikzpicture}[scale=1.5] \tikzset {decoration={snake,amplitude=.6mm,segment length=1.5mm, post length=0mm,pre length=0mm}} \draw [thick] (-0.5,0) -- (0.5,0); \draw [thick,-latex] (0.5,0) -- (0.2,0); \draw [thick,-latex] (0,0) -- (-0.35,0); \fill (0,0) circle[radius=1.2pt]; \draw [thick,decorate] (0,0) -- (0,0.25); \fill (0,0.25) circle[radius=1.2pt]; \draw [thick] (0,0.25) to[out=45,in=135,loop] (0,0.25); \draw (0,-0.5) node{$~$}; \draw [thick,-latex] (0.01,0.51) -- (-0.08,0.51); \end {tikzpicture}} ~+ ~\mathcal {O}(q^4)\quad . \{end}{flalign}$

Evaluating these diagrams with the above Feynman rules, together with the ‘sign rule’ that each closed fermion loop comes with a factor of $-1$, one obtains

\begin{flalign} \nn &\expect {\Omega }{\TO \big (\Psi (x)\,\overline {\Psi }(y)\big )}{\Omega }\,=\,S_{F}(x-y)\\ \nn &\quad \quad + q^2 \int _{\bbR ^{2d}} S_{F}(x-z_1)\gamma ^\mu S_{F}(z_1-z_2)\gamma ^\nu S_{F}(z_2-y)\,(D_{F})_{\mu \nu }(z_1-z_2)~\dd z_1\,\dd z_2 \\ &\quad \quad - q^2 \int _{\bbR ^{2d}} S_{F}(x-z_1)\gamma ^\mu S_{F}(z_1-y) ~\mathrm {Tr}\big (S_{F}(0)\gamma ^\nu \big )~ (D_{F})_{\mu \nu }(z_1-z_2)~\dd z_1\,\dd z_2 ~+~\mathcal {O}(q^4)\quad . \end{flalign} To understand the origin of this sign rule, it is a good exercise to re-derive this result from a more rigorous algebraic perspective using the Gell-Mann and Low reduction formula (4.16) and Wick’s theorem for boson and fermion fields. For the latter one has to pay attention to minus signs that arise from exchanging two fermionic field operators, see our discussion in Section 5.4.

Scattering amplitudes:

A new feature of the scattering amplitudes in QED is that the incoming and outgoing particles are polarized and the fermions also carry an electric charge. More precisely, a state $\ket {k,s,\pm q}$ associated with the Dirac field is labeled by the on-shell momentum $k\in \bbR ^d$, spinor polarization $s$ and charge $\pm q$ of the particle (see (5.84)) and a state $\ket {k,\lambda }$ associated with the photon field is labeled by the on-shell momentum $k\in \bbR ^d$ and covector polarization $\lambda $ of the photon (see (6.38)). The LSZ reduction formula (4.68) can be generalized to such types of particles, but due to time constraints we will not discuss the details, which can be found e.g. in the textbook by Greiner/Reinhardt (Exercise 9.2). The result of going through this tedious work is a list of Feynman rules for scattering amplitudes in QED. The momentum space QED Feynman rules for scattering amplitudes to lowest order perturbation theory (i.e. without loop diagram corrections) read as follows:

• Incoming fermion particle

$(7.19) \{begin}{flalign} \parbox {3.5cm}{ \begin{tikzpicture}[scale=1.5] \begin{feynman} \vertex (i1) at (-1.5,0) {$ \ket {k,s,q} $}; \vertex (a) at (0,0); \diagram *{ (i1) -- [fermion] (a), }; \draw [fill=black] (a) circle(0.4mm); \end {feynman} \end {tikzpicture} } = ~~u^s(\mathbf {k}) \{end}{flalign}$
• Outgoing fermion particle

$(7.20) \{begin}{flalign} \parbox {3.5cm}{ \begin{tikzpicture}[scale=1.5] \begin{feynman} \vertex (f1) at (1.5,0) {$ \bra {k,s,q} $}; \vertex (a) at (0,0); \diagram *{ (a) -- [fermion] (f1), }; \draw [fill=black] (a) circle(0.4mm); \end {feynman} \end {tikzpicture} } = ~~\overline {u^s(\mathbf {k})} \{end}{flalign}$
• Incoming fermion antiparticle

$(7.21) \{begin}{flalign} \parbox {3.5cm}{ \begin{tikzpicture}[scale=1.5] \begin{feynman} \vertex (i1) at (-1.5,0) {$ \ket {k,s,-q} $}; \vertex (a) at (0,0); \diagram *{ (a) -- [fermion] (i1), }; \draw [fill=black] (a) circle(0.4mm); \end {feynman} \end {tikzpicture} } = ~~\overline {v^s(\mathbf {k})} \{end}{flalign}$
• Outgoing fermion antiparticle

$(7.22) \{begin}{flalign} \parbox {3.5cm}{ \begin{tikzpicture}[scale=1.5] \begin{feynman} \vertex (f1) at (1.5,0) {$ \bra {k,s,-q} $}; \vertex (a) at (0,0); \diagram *{ (f1) -- [fermion] (a), }; \draw [fill=black] (a) circle(0.4mm); \end {feynman} \end {tikzpicture} } = ~~v^s(\mathbf {k}) \{end}{flalign}$
• Incoming photon

$(7.23) \{begin}{flalign} \parbox {3.5cm}{ \begin{tikzpicture}[scale=1.5] \begin{feynman} \vertex (i1) at (-1.5,0) {$ \ket {k,\lambda } $}; \vertex (a) at (0,0); \diagram *{ (a) -- [photon] (i1), }; \draw [fill=black] (a) circle(0.4mm); \end {feynman} \end {tikzpicture} } = ~~\epsilon ^\lambda (\mathbf {k}) \{end}{flalign}$
• Outgoing photon

$(7.24) \{begin}{flalign} \parbox {3.5cm}{ \begin{tikzpicture}[scale=1.5] \begin{feynman} \vertex (f1) at (1.5,0) {$ \bra {k,\lambda } $}; \vertex (a) at (0,0); \diagram *{ (a) -- [photon] (f1), }; \draw [fill=black] (a) circle(0.4mm); \end {feynman} \end {tikzpicture} } = ~~\epsilon ^\lambda (\mathbf {k}) \{end}{flalign}$
• Internal fermion line

$(7.25) \{begin}{flalign} \parbox {3cm}{ \begin{tikzpicture}[scale=1.5] \begin{feynman} \vertex (a) at (1.5,0); \vertex (b) at (0,0); \diagram *{ (a) -- [fermion,momentum'=$p$] (b), }; \draw (0,-0.5) node {$~~$}; \end {feynman} \end {tikzpicture} } = ~~\frac {-\ii \,\big ({-}\ii \,\slashed {p} +m\big )}{p^2 + m^2 -\ii \,\epsilon } \{end}{flalign}$
• Internal photon line

$(7.26) \{begin}{flalign} \parbox {3cm}{ \begin{tikzpicture}[scale=1.5] \begin{feynman} \vertex (a) at (1.5,0) {$ \mu $}; \vertex (b) at (0,0) {$ \nu $}; \diagram *{ (a) -- [photon,momentum'=$p$] (b), }; \draw (0,-0.5) node {$~~$}; \end {feynman} \end {tikzpicture} } = ~~\frac {-\ii \,\eta _{\mu \nu }}{p^2 -\ii \,\epsilon } \{end}{flalign}$
• Interaction vertex

$(7.27) \{begin}{flalign} \parbox {3cm}{ \begin{tikzpicture}[scale=1.5] \begin{feynman} \vertex (a) at (-0.75,0.5); \vertex (b) at (0,0); \vertex (c) at (-0.75,-0.5); \vertex (d) at (0.9,0) {$ \mu $}; \diagram *{ (a) -- [fermion,momentum=$p_1$] (b) -- [fermion,rmomentum=$p_2$] (c) , (d) -- [photon,momentum'=$p_3$] (b), }; \end {feynman} \draw [fill=black] (b) circle(0.4mm); \end {tikzpicture} } = ~~q\,\gamma ^\mu ~ (2\pi )^d~\delta (p_1+p_2+p_3) \{end}{flalign}$
• Integration over internal momenta $\int _{\bbR ^d} \frac {\dd p}{(2\pi )^d}$
• A minus sign $-1$ for each fermion line crossing

Example 7.2 (Electron-electron scattering). To develop some physical intuition, let’s call the particle/antiparticle associated with $\Psi $ the electron $e^-$ and, respectively, the positron $e^+$. To lowest order in the coupling constant $q=- e$, the following two Feynman diagrams contribute to $e^-e^- \to e^-e^-$ scattering:

$(7.28) \{begin}{flalign} \parbox {14cm}{ \begin{tikzpicture}[scale=1.5] \begin{feynman} \vertex (i1) at (-1.5,0.6) {$\ket {k_1,s_1,-e}$}; \vertex (i2) at (-1.5,-0.6) {$\ket {k_2,s_2,-e}$}; \vertex (f1) at (1.5,0.6) {$\bra {q_1,r_1,-e}$}; \vertex (f2) at (1.5,-0.6) {$\bra {q_2,r_2,-e}$}; \vertex (a) at (0,0.5); \vertex (b) at (0,-0.5); \diagram *{ (i1) -- [fermion] (a) -- [fermion](f1), (i2) -- [fermion] (b) -- [fermion] (f2), (a) -- [photon,momentum'=$p$] (b), }; \draw [fill=black] (a) circle(0.4mm); \draw [fill=black] (b) circle(0.4mm); \end {feynman} \end {tikzpicture} ~\qquad ~ \begin{tikzpicture}[scale=1.5] \begin{feynman} \vertex (i1) at (-1.5,0.6) {$\ket {k_1,s_1,-e}$}; \vertex (i2) at (-1.5,-0.6) {$\ket {k_2,s_2,-e}$}; \vertex (f1) at (1.5,0.6) {$\bra {q_1,r_1,-e}$}; \vertex (f2) at (1.5,-0.6) {$\bra {q_2,r_2,-e}$}; \vertex (a) at (0,0.5); \vertex (b) at (0,-0.5); \diagram *{ (i1) -- [fermion] (a) -- [fermion](f2), (i2) -- [fermion] (b) -- [fermion] (f1), (a) -- [photon,momentum'=$p$] (b), }; \draw [fill=black] (a) circle(0.4mm); \draw [fill=black] (b) circle(0.4mm); \end {feynman} \end {tikzpicture} } \{end}{flalign}$

Using the Feynman rules, we determine the scattering amplitude
$\seteqnumber{0}{7.}{28}$
\begin{flalign} \nn &\big \langle (q_1,r_1,- e),(q_2,r_2,-e);\mathrm {out} \big \vert (k_1,s_1,-e),(k_2,s_2,-e);\mathrm {in}\big \rangle \\ \nn &\quad =\,e^2\, \int _{\bbR ^d}\bigg ( \overline {u^{r_1}(\mathbf {q}_1)}\gamma ^\mu u^{s_1}(\mathbf {k}_1)~ \frac {-\ii \,\eta _{\mu \nu }}{p^2 -\ii \,\epsilon }~\overline {u^{r_2}(\mathbf {q}_2)}\gamma ^\nu u^{s_2}(\mathbf {k}_2)~(2\pi )^{2d}\,\delta (k_1-q_1-p)\,\delta (k_2-q_2+p) \\ \nn &\quad \quad -\overline {u^{r_2}(\mathbf {q}_2)}\gamma ^\mu u^{s_1}(\mathbf {k}_1)~ \frac {-\ii \,\eta _{\mu \nu }}{p^2 -\ii \,\epsilon }~\overline {u^{r_1}(\mathbf {q}_1)}\gamma ^\nu u^{s_2}(\mathbf {k}_2)~(2\pi )^{2d}\,\delta (k_1-q_2-p)\,\delta (k_2-q_1+p) \bigg )\,\frac {\dd p}{(2\pi )^d}\\ &\quad =\, -\ii \,e^2 \,(2\pi )^d\,\delta (k_1+k_2-q_1-q_2)\,\bigg (\tfrac {\overline {u^{r_1}(\mathbf {q}_1)}\gamma ^\mu u^{s_1}(\mathbf {k}_1) ~\overline {u^{r_2}(\mathbf {q}_2)}\gamma _\mu u^{s_2}(\mathbf {k}_2)}{(k_1-q_1)^2} - \tfrac {\overline {u^{r_2}(\mathbf {q}_2)}\gamma ^\mu u^{s_1}(\mathbf {k}_1)~\overline {u^{r_1}(\mathbf {q}_1)}\gamma _\mu u^{s_2}(\mathbf {k}_2)}{(k_1-q_2)^2} \bigg )\quad . \end{flalign} This scattering amplitude contains a lot of physical details, such as how the strength of electron-electron interaction depends on the spinor polarizations (due to the numerators in the parenthesis) and on the momentum transfers $k_1-q_1$ and $k_1-q_2$ (due to the denominator). These fine details are for the moment not too relevant for us, but I would like to emphasize the following more conceptual observation: The electromagnetic interaction between two electrons is mediated by virtual photons, as one can see by looking at the Feynman diagrams above, and its strength is proportional to the product $(-e)\,(-e) = e^2$ of electric charges. With a more refined analysis, see e.g. the textbook by Nastase (Chapter 24.2), one can show that in the nonrelativistic limit this reproduces the well-known Coulomb potential between electric charges.

Example 7.3 (Electron-positron annihilation). Another important process in QED is the annihilation of an electron/positron pair into a pair of photons. This is typically written as $e^- e^+\to \gamma \gamma $, where $\gamma $ refers to the physical name ‘gamma ray/particle’ for photons and not to a gamma matrix. To lowest order in the coupling constant $q=- e$, the following two Feynman diagrams contribute this scattering:

$(7.30) \{begin}{flalign} \parbox {14cm}{ \begin{tikzpicture}[scale=1.5] \begin{feynman} \vertex (i1) at (-1.5,0.6) {$\ket {k_1,s_1,-e}$}; \vertex (i2) at (-1.5,-0.6) {$\ket {k_2,s_2,e}$}; \vertex (f1) at (1.5,0.6) {$\bra {q_1,\lambda _1}$}; \vertex (f2) at (1.5,-0.6) {$\bra {q_2,\lambda _2}$}; \vertex (a) at (0,0.5); \vertex (b) at (0,-0.5); \diagram *{ (i1) -- [fermion] (a) -- [fermion,momentum'=$p$](b) -- [fermion] (i2), (a) -- [photon] (f1), (b) -- [photon] (f2), }; \draw [fill=black] (a) circle(0.4mm); \draw [fill=black] (b) circle(0.4mm); \end {feynman} \end {tikzpicture} ~\qquad ~ \begin{tikzpicture}[scale=1.5] \begin{feynman} \vertex (i1) at (-1.5,0.6) {$\ket {k_1,s_1,-e}$}; \vertex (i2) at (-1.5,-0.6) {$\ket {k_2,s_2,e}$}; \vertex (f1) at (1.5,0.6) {$\bra {q_1,\lambda _1}$}; \vertex (f2) at (1.5,-0.6) {$\bra {q_2,\lambda _2}$}; \vertex (a) at (0,0.5); \vertex (b) at (0,-0.5); \diagram *{ (i1) -- [fermion] (a) -- [fermion,momentum'=$p$] (b) -- [fermion] (i2), (a) -- [photon] (f2), (b) -- [photon] (f1), }; \draw [fill=black] (a) circle(0.4mm); \draw [fill=black] (b) circle(0.4mm); \end {feynman} \end {tikzpicture} } \{end}{flalign}$

The scattering amplitude can be determined by using the Feynman rules, which I leave as an exercise for you.

7.2 Feynman rules and scattering amplitudes in QED

Time-ordered \(n\)-point functions:

Scattering amplitudes:

Further reading