Introduction to Quantum Electrodynamics

We have learnt the classical electrodynamics before, where the interaction is described by the electromagnetic field $A^\mu=(\Phi,\mathbf A),~ F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu$, and interaction with current $\mathcal{L}_{int} = - j^{\mu}A_{\mu}$. However, this theory doesn’t include the generation and annihilation of particles, as well as the quantum theory.

With the Dirac field $\mathcal L_D=\bar\Psi(i \partial\!\!\!/ -m)\Psi$ and photon field introduced before, we can use the current $j^\mu=\bar\Psi\gamma^\mu\Psi$ to construct the interaction term $\mathcal L_{\mathrm{int}} =-q\bar\Psi\gamma^\mu A_\mu\Psi$ with the electromagnetic coupling constants $q=-e$ intuitively. But we can’t tell the uniqueness of such form and why the conservative charge is exactly the electric one. In this section, we will describe the interaction between Dirac field and photon vector field using the gauge transformation.

Gauge Symmetry and QED Lagrangian

In electrodynamics, we know that the transformation $A_\mu(x)\rightarrow A_\mu(x)+\partial_\mu\alpha(x)$ won’t change the physics field, while global $U(1)$ transformation on Dirac field $\Psi\rightarrow e^{-i\alpha}\Psi$ also keeps the Lagrangian invariant. If we promote the $U(1)$ locally $\Psi\rightarrow e^{-iq\lambda(x)}\Psi$ (the $q$ is for later convenience), then the Lagrangian will yield an additional term $-q \Psi(x) \gamma^{\mu} \partial_{\mu} \lambda(x) \bar \Psi(x)$. To remove the additional term, we define the covariant derivative $D_\mu=\partial_\mu+iqA_\mu$ and replace $\partial_{\mu} \to D_{\mu},~ i\bar\Psi \partial\!\!\!/ \Psi \to i\bar\Psi D\!\!\!/ \Psi$, which is Lorentz-invariant. Actually, the $\lambda(x)$ bridges the local phase transformation of Dirac field with the gauge redundancy of photon field which compensates the additional term, allowing us to construct the covariant derivative to introduce their coupling. With the kinetic term of photon field, we can write the Lagrangian now: $\mathcal{L} = -\frac14 F^{\mu\nu}F_{\mu\nu} + i \bar\Psi \partial \!\!\!/ \Psi - m \bar \Psi \Psi + e \bar \Psi \gamma^{\mu} \Psi A_{\mu}$

Now we call the photon field the “gauge field” $A_{\mu}$ (also called connection on internal space), actually we should construct the dynamics for it specially. The commutator of covariant derivative gives $[D_\mu,D_\nu]=ieQ_iF_{\mu\nu}$, which defines the field strength (also curvature in spacetime) and can be used to construct the Lorentz-invariant kinetic term $\mathcal{L}_{A} = -\frac14 F^{\mu\nu}F_{\mu\nu}$. Notice that the mass term of gauge field $\mathcal L_m=\frac12m^2A_\mu A^\mu$ will violent the gauge invariance, the mass of massive gauge boson like $W^\pm,~Z$ will be acquired through Higgs mechanism.

• It seems that using the gauge transformation to define the gauge field as the interaction field is a math game, but it actually satisfies our pursue of unified theory framework.

What’s more, the charge $q=-e$ for electron here is determined by the normalization of electromagnetic field’s kinetic term $-\frac14 F^{\mu\nu}F_{\mu\nu}$, and also called the electromagnetic coupling strength. In more general cases like quarks, we can write $\mathcal L_{\mathrm{int},i} =-eQ_i A_\mu\bar\Psi_i\gamma^\mu\Psi_i$ with the relative charge $q_i = eQ_i$, like $Q_{e^-} =-1$.

A natural question is: how can we recover classical electromagnetic physics from QFT theory? Let’s take the magnetic and spin interaction for examples. First, we can perform the variation of $A_{\mu}$ on $S[\mathcal{L}_{QED}]$ to get Euler–Lagrange equation $\frac{\partial\mathcal L}{\partial A_\nu} -\partial_\mu \frac{\partial\mathcal L} {\partial(\partial_\mu A_\nu)}=0 \to \partial_{\mu} F^{\mu\nu} = q\bar\Psi\gamma^\nu\Psi = J^{\nu}$, which corresponded to Maxwell equations. Another example is to derive the spin magnetic interaction, we can perform the variation of $\bar \Psi$ to get the Dirac equation with charge $(i\gamma^\mu D_\mu-m)\Psi=0$, and organize it as Schrodinger equation with Hamiltonian $i\partial_t\Psi = \left[ \boldsymbol\alpha\cdot(\mathbf p-q\mathbf A) +\beta m+q\Phi \right]\Psi$, with $\boldsymbol\pi=\mathbf p-q\mathbf A,~ \Psi=e^{-imt} (\varphi, \chi)^T$, we can yield $H_{\mathrm{Pauli}} = \frac{(\mathbf p-q\mathbf A)^2}{2m} +q\Phi -\frac{q}{2m}\boldsymbol\sigma\cdot\mathbf B$, which can further give $g=2$ and Lorentz Force.

Spinor QED Scattering

Previous analysis provides the fermion and photon propagator $S(p)=\frac{- p\!\!\!/ + m}{p^2+m^2-i\epsilon}$ and $D_{\mu\nu}^{\text{F}}(p) =\frac{-ig_{\mu\nu}}{p^2-i\epsilon}$ under Feynman gauge. To derive the vertex factor of interaction term $e \bar \Psi \gamma^{\mu} \Psi A_{\mu}$, we similarly use first-order S-matrix $S^{(1)} = -iq\int d^4x\, \bar\Psi(x)\gamma^\mu\Psi(x)A_\mu(x)$, which then gives $\langle f|S^{(1)}|i\rangle \propto (2\pi)^4\delta^{(4)}(p-p'-k) \bar u(p')(-iq\gamma^\mu)u(p)\epsilon_\mu^*(k)$. Now we can read the vertex factor $ie \gamma^{\mu}$, as well as the particle line — $u_s(p)$ for incoming electron as $\langle0|\Psi(x) b^{\dagger}_s(p)|0\rangle =u_s(p)e^{-ipx}$, $\bar v_s (p)$ for incoming positron, and their conjugate for outgoing ones, like $\langle0|b_s (p) \bar\Psi(x)|0\rangle =\bar u_s(p)e^{ipx}$ at final states; the incoming photon corresponds to $\epsilon_{\mu}^{(\lambda)}(k)$ due to $\langle0|A_\mu(x) a^{\dagger}_{\lambda}(k)|0\rangle =\epsilon_\mu^{(\lambda)}(k)e^{-ikx}$, and outgoing photon is $\epsilon_{\mu}^{(\lambda)*}(k)$.

Now we can calculate the scattering cross section. The first example is the $e^-\mu^-\to e^-\mu^-$ process, the Feynman diagram is shown below, with exchange of a virtual photon. Thus we can use the Feynman Rules to write the amplitude $i\mathcal M= \bar u_e(p_3)(-iq_e\gamma^\mu)u_e(p_1) \times\frac{-ig_{\mu\nu}}{k^2+i\epsilon} \times\bar u_\mu(p_4)(-iq_\mu\gamma^\nu)u_\mu(p_2)$. Considering different spin states, we should average unpolarized initial states while sum up undefined final states as $\overline{|\mathcal M|^2} = \frac14 \sum_{s_1,s_2} \sum_{s_3,s_4} |\mathcal M_{s_3s_4;s_1s_2}|^2$, then $\mathcal M = \frac{e^2}{t} (\bar u_3\gamma^\mu u_1) (\bar u_4\gamma_\mu u_2)$ becomes $|\mathcal M|^2 =\frac{e^4}{t^2} (\bar u_3\gamma^\mu u_1) (\bar u_1\gamma^\nu u_3) \times (\bar u_4\gamma_\mu u_2) (\bar u_2\gamma_\nu u_4)$, then using $\sum_{s_1,s_3} (\bar u_3\gamma^\mu u_1) (\bar u_1\gamma^\nu u_3) = \operatorname{Tr} \left[ (\not p_3+m_e)\gamma^\mu (\not p_1+m_e)\gamma^\nu \right]$, we have $\overline{|\mathcal M|^2} =\frac{2e^4}{t^2} \left[ (s-m_e^2-m_\mu^2)^2 +(u-m_e^2-m_\mu^2)^2 +2t(m_e^2+m_\mu^2) \right]$ and $\frac{d\sigma}{d\Omega} = \frac{1}{64\pi^2s} \frac{|\mathbf p_f|}{|\mathbf p_i|} \overline{|\mathcal M|^2}$.

• Consider non-relativistic situation, using $\bar u_{s'}(p)\gamma^\mu u_s(p)=2p^\mu\delta_{s's}$ where $p_1^{\mu} p_{2\mu} \simeq m_1 m_2$ and $t=k^2\simeq-|\mathbf q|^2$, we can write $\mathcal M_{\mathrm{NR}} \simeq-\frac{4e^2m_em_\mu}{|\mathbf q|^2} \triangleq =-4m_em_\mu\,\widetilde V(\mathbf q)$ under Born approximation, then using Fourier transformation we have $V(r)=\frac{e^2}{4\pi r}=\frac{\alpha}{r}$, the column potential! As $s\simeq(m_e+m_\mu)^2$, we introduce $\mu=\frac{m_em_\mu}{m_e+m_\mu}$ and write $|\mathbf p|=\mu v,~ t\simeq-4\mu^2v^2\sin^2\frac{\theta}{2}$, then derive $\frac{d\sigma}{d\Omega} =\frac{\alpha^2} {4\mu^2v^4\sin^4(\theta/2)}$, which is the same with Rutherford differential cross section. In this way, we have recovered the classical correspondence!

e⁻ ───────────●─────────── e⁻
              ┆ γ
              ┆
μ⁻ ───────────●─────────── μ⁻

Another example is the $e^+ e^- \to \mu^+ \mu^-$ process, where the annihilation of fermions produces the “virtual photon”. We have $i\mathcal M= \left[\bar v(p_2)(ie\gamma^\mu)u(p_1)\right] \times\frac{-ig_{\mu\nu}}{s+i\epsilon} \times\left[\bar u(p_3)(ie\gamma^\nu)v(p_4)\right]$, and then using trace technologies under high-energy approximation $s\gg m_e^2,m_\mu^2$, we get $\overline{|\mathcal M|^2} =\frac{8e^4}{s^2} [ (p_1\cdot p_3)(p_2\cdot p_4) + (p_1\cdot p_4)(p_2\cdot p_3)] \to \overline{|\mathcal M|^2} =e^4(1+\cos^2\theta)$ in the rest frame. This further gives $\frac{d\sigma}{d\Omega} = \frac{\alpha^2}{4s}(1+\cos^2\theta)$ which can be observed in experiment, the QED angular distribution. The virtual photon propagator actually reflects the quantum correlation of the electromagnetic field. Through $\mathcal{L}_{int} = - j^{\mu}A_{\mu}$, the electromagnetic field couples to charged currents, where the dynamics and creation of charged particle can be described by $j^\mu=\bar\Psi\gamma^\mu\Psi$. Since the electromagnetic field can connect different charged particles, it serves as the mediator in the quantum transition. Although it can carry the physics information such as $q^{\mu}$ and spin, it is not an independently observable particle itself, but it can be used for calculation and prediction as an internal contribution.

• We can use Fourier transformation $\int\frac{d^3q}{(2\pi)^3} \frac{e^{i\mathbf q\cdot\mathbf r}} {|\mathbf q|^2} =\frac1{4\pi r}$ to recover the Coulomb potential $V(r)=\frac{q_1q_2}{4\pi r}$ between two charged particles.

All above, we construct our first QFT theory for electromagnetic interaction at tree level! We will further discuss the loop correction, which show the power of QFT prediction. As for decay, we will discuss them in electroweak theory later, like decay $\mu^- \to \nu_{\mu} e^- \bar \nu_{e}$.

Yukawa Theory and Scalar QED

In fact, the Dirac field can also interact with scalar field, which is called the Yukawa Theory. Consider the free Lagrangian $\mathcal L_0=\frac12\partial_\mu\phi\partial^\mu\phi-\frac12M^2\phi^2 + \bar\Psi(i\gamma^\mu\partial_\mu-m)\Psi$ with neutral real scalar field, we can write the Lorentz-invariant Yukawa interaction term $\mathcal{L}_{Yuk} = -y \varphi \bar \Psi \Psi$ with the vertex factor $(-iy)$. Notice that the local $U(1)$ gauge transformation is satisfied here, it doesn’t require the existence of Yukawa interaction like that in spinor QED. Furthermore, the variation on $\bar \Psi$ will yield $(i \partial \!\!\!/ -m-y\phi)\Psi=0$, where we can write $m_{eff}(x) = m + y\varphi(x)$ that the scalar field can influence the dynamics of Dirac field like endowing mass. This will be further discussed in the Higgs mechanism about the mass origin of elementary particles.

Let’s deal with the $\varphi \to \Psi \Psi$ decay process. We have $i\mathcal M_{s_1s_2} =-iy\,\bar u_{s_1}(p_1)v_{s_2}(p_2)$, and derive $\sum_{s}|\mathcal M|^2 = 2y^2(M^2-4m^2)$, then using $d\Gamma=\frac{1}{2M} \sum_{\mathrm{spins}}|\mathcal M|^2\,d\Phi_2$, we get the decay rate $\Gamma(\phi\to f\bar f) = \frac{y^2M}{8\pi} \left(1-\frac{4m^2}{M^2}\right)^{3/2}$, where you can notice that $m \to M/2$ will cause $\Gamma \to 0$ due to reduction in momentum phase space and recognition of the p-wave suppression that $\mathcal M\propto \boldsymbol\sigma\cdot\mathbf p \sim (l=1)$ in $Y(\theta,\phi)_{1,m}$ angular structure. A real example is the Higgs decay to a pair of muons (the second-generation fermion) $H \to \mu^+ \mu^-$. After the breaking of electroweak symmetry, the interaction term can be written as $\mathcal L_{\mathrm{int}} =-\frac{m_\mu}{v}H\bar \Psi_{\mu} \Psi_{\mu}$, which gives $\Gamma(H\to\mu^+\mu^-) \simeq\frac{m_\mu^2m_H}{8\pi v^2}$, which is also called Branch Ratio.

Another process is $\Psi \Psi \to \Psi \Psi$ with exchange of a scalar particle, where we have $i\mathcal M= [\bar u_3(-iy)u_1] \times\frac{i}{k^2-M^2+i\epsilon} \times[\bar u_4(-iy)u_2] = -\frac{y^2}{k^2-M^2} (\bar u_3u_1)(\bar u_4u_2)$. In non-relativistic situation, we have $\bar u(p')u(p) \simeq2m\,\delta_{ss'},~ k^2 \simeq -|\mathbf{q}|^2$, which gives $\mathcal M_{\mathrm{NR}} \simeq \frac{4m^2y^2} {|\mathbf q|^2+M^2} \triangleq \mathcal M_{\mathrm{NR}} =-4m^2 \widetilde V(\mathbf q)$, then we can get $V(r)= \int\frac{d^3q}{(2\pi)^3} e^{i\mathbf q\cdot\mathbf r} \widetilde V(\mathbf q) = -\frac{y^2}{4\pi r}e^{-Mr}$, which is the Yukawa potential that describes the exchange of massive scalar particle in short-distance interaction.

More completely, we can construct the scalar QED that describes the interaction between . As introduced in Section I, the charged scalar particle needs to be represented by complex scalar field, with the covariant derivative $D_\mu=\partial_\mu+iqA_\mu$ we have $\mathcal L_{\mathrm{QED}}^{s} =(D_\mu\phi)^*(D^\mu\phi) -m^2\phi^*\phi -\frac14F_{\mu\nu}F^{\mu\nu}$, which gives the interaction term $\mathcal{L}_{int} = iq A_{\mu} (\varphi \partial^{\mu} \varphi^* - \varphi^* \partial^{\mu} \varphi) + q^2 A_{\mu} A^{\mu} \varphi^* \varphi$. The second-order kinetic term yield 3-point $-iq (p+p')^{\mu}$ and 4-point vertex $2iq^2 g^{\mu\nu}$, which is different from spinor QED. An example process is the $e^+ e^- \to \varphi^+ \varphi^-$ at tree level, where $i\mathcal M = \left[ie\bar v(p_2)\gamma^\mu u(p_1)\right] \times\frac{-ig_{\mu\nu}}{s+i\epsilon} \times\left[ie(p_3-p_4)^\nu\right]$ and gives $\overline{|\mathcal M|^2} = \frac{e^4}{2} \left(1-\frac{4m_\varphi^2}{s}\right) \left[ \sin^2\theta+ \frac{4m_e^2}{s}\cos^2\theta \right]$, then under high-energy approximation $s\gg m_e^2,m_\varphi^2$ we have $\frac{d\sigma}{d\Omega} = \frac{\alpha^2}{8s} \sin^2\theta$ , which is different from $\frac{d\sigma}{d\Omega} (e^+ e^- \to \mu^+ \mu^-) = \frac{\alpha^2}{4s}(1+\cos^2\theta)$. Furthermore, we can derive $\frac{\sigma(e^+ e^- \to \varphi^+ \varphi^-)}{\sigma(e^+ e^- \to \mu^+ \mu^-)} =\frac14$, which is caused by the spin properties and can be used to identify the spin of particles in final states.

All together the Lagrangian of $\varphi,~\phi,~\Psi,~A^{\mu}$ (real scalar field, complex scalar field, Dirac spinor field, photon gauge vector field) can be summarized to describe a parity-invariant extension of QED with Yukawa and scalar interactions. While the parity violation will allow more terms like a pseudoscalar coupling $(-iy_P\varphi\bar\Psi\gamma^5\Psi)$, we will discuss it in later symmetry and electroweak interaction. Note that we don’t include higher-order terms to satisfy later renormalizability, and this theory can provide perturbative prediction with weak coupling while neglecting effects such as the internal structure of composite particles like $\pi^{0,\pm}$. The validity of a physics theory depends on the energy scale and approximations. $\mathcal L= -\frac14F_{\mu\nu}F^{\mu\nu} +\bar\Psi(i\not D_e-m_e)\Psi+(D_\mu\phi)^\dagger(D^\mu\phi) +\frac12(\partial_\mu\varphi)^2-y\varphi\bar\Psi\Psi -V(\varphi,\phi^\dagger\phi)$ $V(\varphi,\phi^\dagger\phi) =V_0+t\varphi +\frac12m_\varphi^2\varphi^2 +\frac{\kappa}{3!}\varphi^3 +\frac{\lambda_\varphi}{4!}\varphi^4+m_\phi^2\phi^\dagger\phi +\mu\varphi\phi^\dagger\phi+\frac{\lambda_{\varphi\phi}}2 \varphi^2\phi^\dagger\phi+\lambda_\phi(\phi^\dagger\phi)^2$