Complete Quantum Field Theory

Part I: Feynman Diagrams and Quantum Electrodynamics

Core ideas

QFT combines quantum mechanics, special relativity, and many-particle physics. Particles are excitations of fields, interactions are local terms in a Lagrangian, and perturbation theory organizes scattering amplitudes as Feynman diagrams.

For review, be able to explain why fields replace fixed-particle wave functions, identify propagators and vertices, and connect amplitudes to observables. Keep the physical question visible: identify the degrees of freedom, the conserved quantities, the approximation being made, and the observable that would be measured.

Mathematical spine

$$S=\int d^4x\,\mathcal L,\qquad \langle f|S|i\rangle=\delta_{fi}+i(2\pi)^4\delta^4(P_f-P_i)\mathcal M$$

Worked example

The free real scalar field has Lagrangian $\mathcal L=\frac12(\partial_\mu\phi)^2-\frac12m^2\phi^2$. The equation of motion is $(\Box+m^2)\phi=0$. For a plane wave $\phi(x)=e^{-ik\cdot x}$, this gives $k^2=m^2$, i.e., $E^2=\bm p^2+m^2$. The on-shell condition enforces the relativistic energy-momentum relation.

Problems with Solutions

Problem 1. Show that the free scalar Lagrangian is invariant under the global phase transformation $\phi\to e^{i\alpha}\phi$. Solution. For a real scalar field, $\phi$ does not transform by a phase. For a complex scalar, $\phi^*\phi$ is invariant. Since $\mathcal L$ depends only on $\phi^2$ and $(\partial_\mu\phi)^2$, and both are invariant under $\phi\to-\phi$, the Lagrangian has a discrete $\mathbb Z_2$ symmetry, not a continuous U(1).

Problem 2. What are the dimensions of the scalar field $\phi$ in $d=4$ spacetime dimensions? Solution. The kinetic term $(\partial\phi)^2$ has dimension 4, so $[\partial]+[\phi]=2$, giving $[\phi]=1$ (mass dimension). The mass term $m^2\phi^2$ then has dimension $2+2=4$, consistent.

Section summary. QFT computes relativistic quantum processes through fields and amplitudes.

The Klein-Gordon Field

Core ideas

The Klein-Gordon field is the simplest relativistic quantum field. Quantization turns each momentum mode into a harmonic oscillator and introduces creation and annihilation operators. The propagator is the Green function for relativistic propagation.

For review, be able to derive the equation of motion, quantize modes, interpret particles and antiparticles, and write the scalar propagator. Keep the physical question visible: identify the degrees of freedom, the conserved quantities, the approximation being made, and the observable that would be measured.

Mathematical spine

$$\mathcal L=\frac12\partial_\mu\phi\,\partial^\mu\phi-\frac12m^2\phi^2,\qquad (\Box+m^2)\phi=0,\qquad \Delta_F(p)=\frac{i}{p^2-m^2+i\epsilon}$$

Worked example

For a scalar field of mass $m=125\,\mathrm{GeV}$ (the Higgs boson), the Feynman propagator at momentum $p^\mu=(E,\bm 0)$ is $\Delta_F(E)=i/(E^2-m^2+i\epsilon)$. At $E=200\,\mathrm{GeV}$, $|\Delta_F|\sim1/(200^2-125^2)\approx1/(2.34\times10^4)\approx4.3\times10^{-5}\,\mathrm{GeV^{-2}}$. Near resonance $E\approx m$, the propagator blows up as $i/(2m(E-m)+i\epsilon)$, producing the characteristic Breit-Wigner peak.

Problems with Solutions

Problem 1. Compute the equal-time commutator $[\phi(\bm x,t),\pi(\bm y,t)]$ for the free Klein-Gordon field. Solution. Canonical quantization gives $[\phi(\bm x,t),\pi(\bm y,t)]=i\delta^{(3)}(\bm x-\bm y)$, which is the field-theoretic analogue of $[q_i,p_j]=i\delta_{ij}$.

Problem 2. A real scalar field has Fourier expansion $\phi(x)=\int\frac{d^3p}{(2\pi)^3}\frac{1}{\sqrt{2E_{\bm p}}}(a_{\bm p}e^{-ip\cdot x}+a_{\bm p}^\dagger e^{ip\cdot x})$. Show that $a_{\bm p}^\dagger$ creates a particle of momentum $\bm p$. Solution. The Hamiltonian is $H=\int\frac{d^3p}{(2\pi)^3}E_{\bm p}a_{\bm p}^\dagger a_{\bm p}$. Acting on the vacuum: $H a_{\bm p}^\dagger|0\rangle=E_{\bm p}a_{\bm p}^\dagger|0\rangle$, confirming $a_{\bm p}^\dagger|0\rangle$ is a one-particle state of energy $E_{\bm p}=\sqrt{\bm p^2+m^2}$.

Section summary. Scalar fields show how particles emerge from quantized modes.

The Dirac Field

Core ideas

The Dirac field describes spin-$1/2$ fermions. Gamma matrices linearize the relativistic dispersion relation, spinors carry Lorentz representation data, and anticommutation enforces Pauli exclusion and positive energy.

For review, be able to use the Dirac equation, spinor completeness, conserved current, and the fermion propagator. Keep the physical question visible: identify the degrees of freedom, the conserved quantities, the approximation being made, and the observable that would be measured.

Mathematical spine

$$\mathcal L=\bar\psi(i\gamma^\mu\partial_\mu-m)\psi,\qquad (i\gamma^\mu\partial_\mu-m)\psi=0,\qquad S_F(p)=\frac{i(\not p+m)}{p^2-m^2+i\epsilon}$$

Worked example

For an electron at rest ($\bm p=0$), the Dirac equation reduces to $i\gamma^0\partial_0\psi=m\psi$. In the Dirac representation, $\gamma^0=\begin{pmatrix}I&0\\0&-I\end{pmatrix}$, so the positive-energy solution has upper two components $\psi_+\sim e^{-imt}u(0)$ and the negative-energy solution has lower components $\psi_-\sim e^{+imt}v(0)$. The existence of negative-energy solutions requires the interpretation of antiparticles (holes in the Dirac sea).

Problems with Solutions

Problem 1. Show that $\{\gamma^\mu,\gamma^\nu\}=2g^{\mu\nu}$ implies $(\not p)^2=p^2$. Solution. $(\not p)^2=\gamma^\mu\gamma^\nu p_\mu p_\nu=\frac12\{\gamma^\mu,\gamma^\nu\}p_\mu p_\nu=g^{\mu\nu}p_\mu p_\nu=p^2$.

Problem 2. Compute the trace $\mathrm{Tr}(\gamma^\mu\gamma^\nu)$. Solution. Using $\gamma^\mu\gamma^\nu+\gamma^\nu\gamma^\mu=2g^{\mu\nu}$ and taking the trace: $2\mathrm{Tr}(\gamma^\mu\gamma^\nu)=2g^{\mu\nu}\mathrm{Tr}(I)=8g^{\mu\nu}$ in 4D (since $\mathrm{Tr}(I)=4$). Thus $\mathrm{Tr}(\gamma^\mu\gamma^\nu)=4g^{\mu\nu}$.

Section summary. Dirac fields are relativistic quantum fields for fermions.

Interacting Fields and Feynman Diagrams

Core ideas

Interactions make fields scatter and decay. Perturbation theory expands time-ordered correlation functions in coupling constants; Wick’s theorem reduces products of free fields to propagators and vertices.

For review, be able to derive simple Feynman rules, count symmetry factors, and relate diagrams to powers of coupling. Keep the physical question visible: identify the degrees of freedom, the conserved quantities, the approximation being made, and the observable that would be measured.

Mathematical spine

$$\mathcal L=\mathcal L_0+\mathcal L_{\rm int},\qquad Z[J]=\int\mathcal D\phi\,e^{iS[\phi]+i\int J\phi}$$

Worked example

Consider $\phi^4$ theory with $\mathcal L_{\rm int}=-\frac{\lambda}{4!}\phi^4$. The first-order correction to the two-point function $\langle\phi(x)\phi(y)\rangle$ comes from the diagram with one vertex connected to two external lines by propagators. The symmetry factor for this “tadpole” diagram is $4!/2=12$, but the two external lines can attach in $4\times3=12$ ways, giving a factor of $\lambda/2$. The corrected propagator is $\Delta_F(x-y)-\frac{i\lambda}{2}\Delta_F(0)\int d^4z\,\Delta_F(x-z)\Delta_F(z-y)$.

Problems with Solutions

Problem 1. In $\phi^4$ theory, how many distinct ways can four external lines attach to a single $\phi^4$ vertex? Solution. The vertex has 4 identical legs. The number of ways to choose 4 external lines (with replacement, order matters) is $4!=24$, but permutations of the legs give the same diagram. The symmetry factor for the 4-point function at tree level is $4!/4!=1$, so there is exactly one tree-level diagram.

Problem 2. Write Wick’s theorem for $\langle\phi_1\phi_2\phi_3\phi_4\rangle$ in terms of two-point functions. Solution. $\langle\phi_1\phi_2\phi_3\phi_4\rangle=\langle\phi_1\phi_2\rangle\langle\phi_3\phi_4\rangle+\langle\phi_1\phi_3\rangle\langle\phi_2\phi_4\rangle+\langle\phi_1\phi_4\rangle\langle\phi_2\phi_3\rangle$.

Section summary. Feynman diagrams are bookkeeping for perturbative correlation functions.

Elementary Processes of QED

Core ideas

QED couples the Dirac field to the electromagnetic gauge field. Tree-level processes such as electron-muon scattering, annihilation, pair production, and Compton scattering illustrate spinor algebra, gauge invariance, and cross-section calculations.

For review, be able to write QED Feynman rules, use Ward identities conceptually, and compute the structure of basic amplitudes. Keep the physical question visible: identify the degrees of freedom, the conserved quantities, the approximation being made, and the observable that would be measured.

Mathematical spine

$$\mathcal L_{\rm QED}=-\frac14F_{\mu\nu}F^{\mu\nu}+\bar\psi(i\gamma^\mu D_\mu-m)\psi,\qquad D_\mu=\partial_\mu+ieA_\mu$$

Worked example

The QED vertex factor is $-ie\gamma^\mu$. For electron-muon scattering $e^-\mu^-\to e^-\mu^-$, the tree-level amplitude is $\mathcal M=(-ie)^2(\bar u_{e'}\gamma^\mu u_e)(-ig_{\mu\nu}/q^2)(\bar u_{\mu'}\gamma^\nu u_\mu)$. In the non-relativistic limit this reduces to the Coulomb scattering amplitude, with the spinor structure giving the correct magnetic moment coupling.

Problems with Solutions

Problem 1. Write the Feynman amplitude for Compton scattering $\gamma+e^-\to\gamma+e^-$ at tree level. Solution. There are two diagrams: (s-channel) $\mathcal M_s=-e^2\bar u_{e'}\slashed\epsilon'(\not p+\not k+m)\slashed\epsilon u_e/((p+k)^2-m^2)$ and (u-channel) with $k\leftrightarrow -k'$. The sum gives the Klein-Nishina formula.

Problem 2. What is the Ward identity for the QED 3-point function? Solution. $k_\mu\mathcal M^\mu=0$, where $k_\mu$ is the photon momentum. This ensures gauge invariance and guarantees that the photon remains massless to all orders.

Section summary. QED is the cleanest example of a quantum gauge theory.

Radiative Corrections

Core ideas

Loops correct masses, charges, magnetic moments, and scattering amplitudes. They introduce divergences that must be regularized and renormalized. Physical predictions depend on measured parameters at a scale, not on bare quantities.

For review, be able to identify self-energy, vertex, and vacuum polarization corrections, explain regularization, and interpret running charge. Keep the physical question visible: identify the degrees of freedom, the conserved quantities, the approximation being made, and the observable that would be measured.

Mathematical spine

$$\alpha(q^2)=\frac{\alpha(\mu^2)}{1-\Pi(q^2,\mu^2)},\qquad g-2\ \mathrm{starts\ at}\ \frac{\alpha}{2\pi}$$

Worked example

The electron anomalous magnetic moment is $a_e=(g-2)/2$. Schwinger’s 1948 one-loop calculation gave $a_e^{(2)}=\alpha/(2\pi)\approx0.00116$. Using $\alpha\approx1/137.036$, $a_e^{(2)}\approx0.0011614$. The current best experimental value is $a_e^{\exp}=0.00115965218059(13)$, matching QED predictions to 10 significant figures after including four-loop corrections.

Problems with Solutions

Problem 1. Estimate the one-loop vacuum polarization contribution to the running of $\alpha$ at $Q^2=m_Z^2$. Solution. The one-loop result is $\Pi(Q^2)=-\frac{\alpha}{3\pi}\ln(Q^2/m_e^2)$ for $Q^2\gg m_e^2$. At $Q=m_Z$, $\Pi\approx-(1/137)/(3\pi)\ln((91\,\mathrm{GeV})^2/(0.511\,\mathrm{MeV})^2)\approx-0.00077\times\ln(3.2\times10^{10})\approx-0.018$. Thus $\alpha(m_Z^2)\approx\alpha(0)/(1+0.018)\approx1/128$.

Problem 2. Why must the photon self-energy $\Pi^{\mu\nu}(q)$ be transverse, $q_\mu\Pi^{\mu\nu}=0$? Solution. Gauge invariance (Ward identity) requires that the photon remain massless. A massive photon would correspond to a pole at $q^2\neq0$, but the transversality condition ensures $\Pi^{\mu\nu}(q)=(g^{\mu\nu}q^2-q^\mu q^\nu)\Pi(q^2)$, with no mass term.

Section summary. Radiative corrections make QFT predictive beyond tree level.

Part II: Renormalization

Core ideas

Renormalization separates short-distance unknowns from long-distance predictions. Counterterms absorb divergences, while renormalized couplings depend on scale. Relevant, marginal, and irrelevant operators explain why low-energy physics can be universal.

For review, be able to classify operators by dimension, state what counterterms do, and distinguish bare from renormalized parameters. Keep the physical question visible: identify the degrees of freedom, the conserved quantities, the approximation being made, and the observable that would be measured.

Mathematical spine

$$\mathcal L=\mathcal L_{\rm ren}+\mathcal L_{\rm counter},\qquad \mu\frac{dg}{d\mu}=\beta(g)$$

Worked example

In $\phi^4$ theory in $d=4$, the bare coupling $\lambda_0$ is related to the renormalized coupling $\lambda$ by $\lambda_0=\mu^\epsilon(\lambda+\delta_\lambda)$ in dimensional regularization ($d=4-\epsilon$). At one loop, $\delta_\lambda=-3\lambda^2/(16\pi^2\epsilon)$. The beta function is $\beta(\lambda)=3\lambda^2/(16\pi^2)+O(\lambda^3)$, showing that the coupling grows with energy ( Landau pole).

Problems with Solutions

Problem 1. In QED, the one-loop beta function is $\beta(e)=e^3/(12\pi^2)$. Solve for the running coupling $e(\mu)$. Solution. $\frac{de}{d\ln\mu}=\frac{e^3}{12\pi^2}$. Integrating: $-\frac{1}{2e^2}\big|_{e_0}^{e}=\frac{1}{12\pi^2}\ln(\mu/\mu_0)$, giving $\frac{1}{e^2(\mu)}=\frac{1}{e_0^2}-\frac{1}{6\pi^2}\ln(\mu/\mu_0)$. The coupling increases with energy.

Problem 2. What is the superficial degree of divergence of a QED diagram with $V$ vertices, $I_e$ internal electron lines, and $I_\gamma$ internal photon lines? Solution. $D=4L-2I_\gamma-I_e$ where $L=I_e+I_\gamma-V+1$. Using $2I_e+E_e=2V$ and $2I_\gamma+E_\gamma=V$, one finds $D=4-E_\gamma-\frac32E_e$, showing that only a finite number of amplitudes diverge.

Section summary. Renormalization is scale-dependent bookkeeping of physical parameters.

Functional Methods

Core ideas

Path integrals compute generating functionals for correlation functions. Sources generate expectation values, effective actions generate one-particle-irreducible vertices, and saddle points connect quantum field theory to classical field equations.

For review, be able to use generating functionals, functional derivatives, Wick rotation, and effective action concepts. Keep the physical question visible: identify the degrees of freedom, the conserved quantities, the approximation being made, and the observable that would be measured.

Mathematical spine

$$Z[J]=\int\mathcal D\phi\,\exp\left(iS[\phi]+i\int J\phi\,d^4x\right),\qquad \langle\phi(x)\phi(y)\rangle=\frac{1}{i^2Z}\frac{\delta^2Z}{\delta J(x)\delta J(y)}$$

Worked example

For a free scalar field, the generating functional is $Z_0[J]=Z_0[0]\exp(-\frac12\int d^4x\,d^4y\,J(x)\Delta_F(x-y)J(y))$. The two-point function is $\langle\phi(x)\phi(y)\rangle=\Delta_F(x-y)$, the Feynman propagator. This follows from taking two functional derivatives of $Z_0[J]$ with respect to $J$ and setting $J=0$.

Problems with Solutions

Problem 1. Compute the four-point function $\langle\phi(x_1)\phi(x_2)\phi(x_3)\phi(x_4)\rangle$ for the free scalar field from the generating functional. Solution. Taking four functional derivatives of $Z_0[J]$ gives the sum over all pairings: $\Delta_F(x_1-x_2)\Delta_F(x_3-x_4)+\Delta_F(x_1-x_3)\Delta_F(x_2-x_4)+\Delta_F(x_1-x_4)\Delta_F(x_2-x_3)$, which is Wick’s theorem.

Problem 2. What is the effective action $\Gamma[\phi_{\rm cl}]$ and how does it relate to 1PI diagrams? Solution. $\Gamma[\phi_{\rm cl}]$ is the Legendre transform of $W[J]=-i\ln Z[J]$. Its expansion in $\phi_{\rm cl}$ generates all one-particle-irreducible (1PI) vertex functions, which are the building blocks of Feynman diagrams.

Section summary. Functional methods make symmetries and correlation functions systematic.

Systematics of Renormalization

Core ideas

Renormalization requires all counterterms allowed by symmetries. Power counting tells which diagrams diverge; schemes such as dimensional regularization and minimal subtraction define finite parts. Symmetry identities constrain counterterms.

For review, be able to perform superficial degree-of-divergence counting, explain schemes, and use symmetry to restrict terms. Keep the physical question visible: identify the degrees of freedom, the conserved quantities, the approximation being made, and the observable that would be measured.

Mathematical spine

$$D=4L-2I_B-I_F+\sum_v d_v,\qquad \frac{1}{\epsilon}\ \mathrm{poles\ are\ absorbed\ into\ counterterms}$$

Worked example

Consider a scalar theory with interaction $\mathcal L_{\rm int}=-g\phi^3$ in $d=4$. The superficial degree of divergence of a diagram with $L$ loops, $I$ internal lines, and $V$ vertices is $D=4L-2I+3V$. Using $L=I-V+1$ and $3V=2I+E$ (each vertex has 3 lines, $E$ external), we get $D=4-E$. Thus only the 2-point ($E=2$, $D=2$) and 3-point ($E=3$, $D=1$) functions are superficially divergent, requiring mass, field-strength, and coupling counterterms.

Problems with Solutions

Problem 1. In $\phi^4$ theory in $d=4$, which correlation functions are superficially divergent? Solution. $D=4-E$, so only $E=2$ ($D=2$) and $E=4$ ($D=0$) are divergent. This means only the propagator and 4-point vertex need renormalization.

Problem 2. Why does dimensional regularization preserve gauge invariance? Solution. Because the regulator $\epsilon=4-d$ does not introduce any mass scale or violate the symmetry. The integration measure and Feynman rules are analytically continued in $d$, maintaining the Ward identities at every step.

Section summary. The structure of divergences is organized by dimension and symmetry.

Renormalization Group

Core ideas

The renormalization group describes how theories change with scale. Fixed points, beta functions, anomalous dimensions, and running couplings explain asymptotic freedom, critical phenomena, and effective field theory.

For review, be able to interpret beta functions, fixed points, relevant perturbations, and running couplings. Keep the physical question visible: identify the degrees of freedom, the conserved quantities, the approximation being made, and the observable that would be measured.

Mathematical spine

$$\mu\frac{dg_i}{d\mu}=\beta_i(\{g\}),\qquad \mathcal O(\lambda x)=\lambda^{-\Delta}\mathcal O(x)$$

Worked example

In $\phi^4$ theory, $\beta(\lambda)=3\lambda^2/(16\pi^2)$. Integrating from $\mu_0$ to $\mu$: $\lambda(\mu)=\lambda(\mu_0)/(1-\frac{3\lambda(\mu_0)}{16\pi^2}\ln(\mu/\mu_0))$. As $\mu\to\infty$, the denominator vanishes at the Landau pole $\mu_L=\mu_0\exp(16\pi^2/(3\lambda(\mu_0)))$. For $\lambda(\mu_0)\sim0.1$, $\mu_L\sim\mu_0\exp(526)$, exponentially far above any physical scale.

Problems with Solutions

Problem 1. QCD has $\beta(g)=-\frac{g^3}{16\pi^2}(11-\frac23n_f)$. For $n_f=3$, find the running coupling at $\mu=10\,\mathrm{GeV}$ if $\alpha_s(1\,\mathrm{GeV})=0.35$. Solution. With $b_0=11-2=9$, $\frac{1}{\alpha_s(\mu)}=\frac{1}{\alpha_s(\mu_0)}+\frac{b_0}{2\pi}\ln(\mu/\mu_0)=2.86+\frac{9}{2\pi}\ln(10)=2.86+3.29=6.15$. Thus $\alpha_s(10\,\mathrm{GeV})\approx0.163$.

Problem 2. What is the anomalous dimension of the scalar field in $\phi^4$ theory at one loop? Solution. The field renormalization gives $Z_\phi=1-\frac{\lambda^2}{12(16\pi^2)^2\epsilon}+\dots$, so the anomalous dimension is $\gamma_\phi=\frac12\mu\frac{d\ln Z_\phi}{d\mu}=O(\lambda^2)$. At one loop in coupling, $\gamma_\phi=0$.

Section summary. RG flow is the map of physics across length scales.

Critical Phenomena

Core ideas

QFT methods apply to continuous phase transitions because long-wavelength fluctuations dominate near criticality. Universality classes depend on symmetry, dimension, and order-parameter components rather than microscopic details.

For review, be able to connect Landau-Ginzburg fields to critical exponents, use correlation length, and explain universality. Keep the physical question visible: identify the degrees of freedom, the conserved quantities, the approximation being made, and the observable that would be measured.

Mathematical spine

$$\mathcal L=\frac12(\partial\phi)^2+\frac12r\phi^2+\frac{u}{4!}\phi^4,\qquad \xi\sim |T-T_c|^{-\nu}$$

Worked example

For the Ising model in $d=3$, the Landau-Ginzburg theory predicts mean-field exponents: $\beta=1/2$, $\gamma=1$, $\nu=1/2$. However, experiments give $\beta\approx0.326$, $\gamma\approx1.24$, $\nu\approx0.63$. The discrepancy arises because mean field theory neglects fluctuations. RG calculations in $4-\epsilon$ dimensions give $\nu\approx1/2+\epsilon/12+\dots$, and for $\epsilon=1$ this is already closer to the observed value.

Problems with Solutions

Problem 1. In mean-field theory, the correlation length diverges as $\xi\sim|r|^{-1/2}$ where $r\sim T-T_c$. What is the critical exponent $\nu$? Solution. Since $\xi\sim|r|^{-\nu}$ and $r\sim T-T_c$, we have $\nu=1/2$ in mean-field theory.

Problem 2. Estimate the upper critical dimension for the Ising model using power counting. Solution. The Gaussian fixed point is stable when the quartic coupling $u$ is irrelevant. In $d$ dimensions, $[u]=4-d$. Thus $u$ is irrelevant for $d>4$, making $d_c=4$ the upper critical dimension above which mean-field theory is exact.

Section summary. Critical phenomena are statistical field theories near fixed points.

Part III: Non-Abelian Gauge Theories

Core ideas

Non-Abelian gauge theories generalize electromagnetism by making gauge fields carry the charge they mediate. This self-interaction is responsible for asymptotic freedom, confinement, and the structure of the Standard Model.

For review, be able to explain local gauge symmetry, covariant derivatives, field strengths, and gauge-boson self-interactions. Keep the physical question visible: identify the degrees of freedom, the conserved quantities, the approximation being made, and the observable that would be measured.

Mathematical spine

$$D_\mu=\partial_\mu-igA_\mu^aT^a,\qquad F_{\mu\nu}^a=\partial_\mu A_\nu^a-\partial_\nu A_\mu^a+gf^{abc}A_\mu^bA_\nu^c$$

Worked example

In $SU(2)$ Yang-Mills theory with generators $T^a=\sigma^a/2$, the structure constants are $f^{abc}=\epsilon^{abc}$. The field strength $F_{\mu\nu}^a=\partial_\mu A_\nu^a-\partial_\nu A_\mu^a+g\epsilon^{abc}A_\mu^bA_\nu^c$ contains a quadratic term in the gauge fields, unlike electromagnetism. This self-interaction is responsible for asymptotic freedom and confinement.

Problems with Solutions

Problem 1. Show that the covariant derivative transforms as $D_\mu\psi\to U(D_\mu\psi)$ under a local gauge transformation $U(x)$. Solution. $D_\mu\psi\to(\partial_\mu+igA'_\mu)U\psi=U\partial_\mu\psi+(\partial_\mu U)\psi+igA'_\mu U\psi$. With $A'_\mu=UA_\mu U^\dagger+\frac{i}{g}(\partial_\mu U)U^\dagger$, this becomes $U(\partial_\mu+igA_\mu)\psi=U(D_\mu\psi)$.

Problem 2. How many physical gluon polarizations exist in $SU(3)$ QCD? Solution. There are 8 gauge bosons ($N^2-1=8$ for $SU(N)$ with $N=3$). Each massless gauge boson has 2 physical polarizations (transverse). Thus there are $8\times2=16$ physical polarization states.

Section summary. Non-Abelian gauge symmetry is the backbone of modern particle physics.

Yang-Mills Theory

Core ideas

Pure Yang-Mills theory contains only non-Abelian gauge fields. Gauge fixing and ghosts are needed for covariant quantization. The theory is classically simple but quantum mechanically rich, with confinement and topological sectors.

For review, be able to write the Yang-Mills action, explain gauge fixing, ghosts, and why gauge redundancy is not a physical degree of freedom. Keep the physical question visible: identify the degrees of freedom, the conserved quantities, the approximation being made, and the observable that would be measured.

Mathematical spine

$$\mathcal L_{\rm YM}=-\frac14F_{\mu\nu}^aF^{a\mu\nu},\qquad \mathcal L_{\rm gf}=-\frac{1}{2\xi}(\partial_\mu A^{a\mu})^2$$

Worked example

The Faddeev-Popov determinant for $SU(2)$ Yang-Mills theory in covariant gauge gives ghost fields $c^a$ with Lagrangian $\mathcal L_{\rm ghost}=\bar c^a(-\partial^2\delta^{ab}-g\epsilon^{abc}\partial^\mu A_\mu^c)c^b$. The ghosts are complex scalars that couple to gauge fields but obey anticommutation relations. They cancel the unphysical longitudinal and timelike polarizations of the gauge bosons in loop diagrams.

Problems with Solutions

Problem 1. What is the gauge-fixing parameter $\xi$ and how does it affect the gluon propagator? Solution. The gluon propagator in covariant gauge is $\Delta_{\mu\nu}^{ab}(q)=\frac{-i\delta^{ab}}{q^2+i\epsilon}\left(g_{\mu\nu}-(1-\xi)\frac{q_\mu q_\nu}{q^2}\right)$. For $\xi=1$ (Feynman gauge) the propagator is simplest; for $\xi=0$ (Landau gauge) it is transverse.

Problem 2. Why are Faddeev-Popov ghosts necessary in non-Abelian gauge theories but not in QED? Solution. In QED the gauge transformation is Abelian and the Faddeev-Popov determinant is field-independent, so it contributes only an overall constant. In non-Abelian theories the determinant depends on $A_\mu$, requiring dynamical ghost fields to maintain unitarity.

Section summary. Yang-Mills theory is the core non-Abelian gauge field theory.

QCD

Core ideas

QCD is Yang-Mills theory with quarks in the fundamental representation of color. It is weakly coupled at high energy and strongly coupled at low energy. Chiral symmetry, confinement, jets, and hadrons are all consequences of color dynamics.

For review, be able to state QCD Lagrangian, explain asymptotic freedom and confinement, and connect quarks/gluons to hadrons and jets. Keep the physical question visible: identify the degrees of freedom, the conserved quantities, the approximation being made, and the observable that would be measured.

Mathematical spine

$$\mathcal L_{\rm QCD}=-\frac14G_{\mu\nu}^aG^{a\mu\nu}+\bar q(i\gamma^\mu D_\mu-m)q,\qquad \beta(g)<0$$

Worked example

At one loop, the QCD beta function is $\beta(g)=-\frac{g^3}{16\pi^2}(11-\frac23n_f)$. For $n_f=3$ light quarks, the coefficient is $b_0=9$. The running coupling is $\alpha_s(Q^2)=\frac{4\pi}{b_0\ln(Q^2/\Lambda_{\rm QCD}^2)}$. With $\Lambda_{\rm QCD}\approx200\,\mathrm{MeV}$, at $Q=2\,\mathrm{GeV}$ we get $\alpha_s=4\pi/(9\ln(4/0.04))=4\pi/(9\times4.6)\approx0.30$, consistent with lattice and experimental values.

Problems with Solutions

Problem 1. At what energy scale does QCD become strongly coupled if $\Lambda_{\rm QCD}=200\,\mathrm{MeV}$? Solution. By definition, $\alpha_s\to\infty$ as $Q\to\Lambda_{\rm QCD}$. The perturbative formula breaks down near $Q\sim\Lambda_{\rm QCD}$, which sets the confinement scale. Hadrons have sizes $R\sim\hbar c/\Lambda_{\rm QCD}\sim1\,\mathrm{fm}$.

Problem 2. Why does asymptotic freedom imply that quarks behave as free particles at short distances? Solution. As $Q^2\to\infty$, $\alpha_s(Q^2)\to0$. In high-energy scattering, the relevant distance scale is $r\sim\hbar/Q$, so at short distances the coupling is weak. This justifies the parton model and perturbative calculations for jets.

Section summary. QCD is the quantum field theory of strong interactions.

Electroweak Theory

Core ideas

Electroweak theory unifies weak and electromagnetic interactions through $SU(2)_L\times U(1)_Y$ gauge symmetry. The Higgs mechanism breaks it to electromagnetism, giving masses to $W$, $Z$, and fermions while leaving the photon massless.

For review, be able to write the symmetry breaking pattern, identify charged and neutral currents, and explain masses from the Higgs vacuum expectation value. Keep the physical question visible: identify the degrees of freedom, the conserved quantities, the approximation being made, and the observable that would be measured.

Mathematical spine

$$SU(2)_L\times U(1)_Y\to U(1)_{\rm em},\qquad M_W=\frac12gv,\qquad M_Z=\frac12\sqrt{g^2+g'^2}\,v$$

Worked example

With measured values $M_W=80.4\,\mathrm{GeV}$ and $M_Z=91.2\,\mathrm{GeV}$, the weak mixing angle is $\cos\theta_W=M_W/M_Z=0.881$, giving $\sin^2\theta_W=1-0.777=0.223$. The $SU(2)_L$ coupling is $g=2M_W/v=2(80.4)/246=0.654$, and the $U(1)_Y$ coupling is $g'=g\tan\theta_W=0.654(0.487)=0.318$. The electric charge is $e=gg'/\sqrt{g^2+g'^2}=0.313$, or $\alpha=e^2/(4\pi)\approx1/128$ at the weak scale.

Problems with Solutions

Problem 1. The Higgs vacuum expectation value is $v=(\sqrt{2}G_F)^{-1/2}\approx246\,\mathrm{GeV}$. Derive this from the muon lifetime formula. Solution. Fermi theory gives $G_F/\sqrt{2}=g^2/(8M_W^2)$. Using $M_W=gv/2$, we get $G_F/\sqrt{2}=1/(2v^2)$, so $v=(\sqrt{2}G_F)^{-1/2}$. With $G_F=1.166\times10^{-5}\,\mathrm{GeV^{-2}}$, $v=(\sqrt{2}\times1.166\times10^{-5})^{-1/2}\approx246\,\mathrm{GeV}$.

Problem 2. How many Goldstone bosons are eaten in electroweak symmetry breaking? Solution. The gauge group $SU(2)_L\times U(1)_Y$ has 4 generators. The unbroken $U(1)_{\rm em}$ has 1 generator. Thus $4-1=3$ Goldstone bosons are eaten by the $W^\pm$ and $Z$ bosons, giving them longitudinal polarizations. The photon remains massless.

Section summary. Electroweak theory is chiral gauge theory with spontaneous symmetry breaking.

Anomalies and Beyond

Core ideas

Anomalies occur when a classical symmetry fails after quantization. Gauge anomalies must cancel for consistency; global anomalies can have physical consequences. Effective field theory extends the Standard Model by higher-dimension operators.

For review, be able to distinguish global and gauge anomalies, state anomaly cancellation, and use EFT as a language for new physics. Keep the physical question visible: identify the degrees of freedom, the conserved quantities, the approximation being made, and the observable that would be measured.

Mathematical spine

$$\partial_\mu j_5^\mu=\frac{g^2}{16\pi^2}F_{\mu\nu}\tilde F^{\mu\nu},\qquad \mathcal L_{\rm EFT}=\mathcal L_{\rm SM}+\sum_i\frac{c_i}{\Lambda^{d_i-4}}\mathcal O_i$$

Worked example

The chiral anomaly in QED with one fermion gives $\partial_\mu j_5^\mu=\frac{e^2}{8\pi^2}\bm E\cdot\bm B$. In a constant background with $E=10^5\,\mathrm{V/m}$ and $B=1\,\mathrm T$, the anomaly density is $\frac{e^2}{8\pi^2\hbar^2c^3}EB\approx\frac{(1.6\times10^{-19})^2}{8\pi^2(1.05\times10^{-34})^2(3\times10^8)^3}(10^5)(1)\approx2.9\times10^{33}\,\mathrm{m^{-3}s^{-1}}$. This enormous rate reflects that the anomaly is a UV effect, not a physical particle production rate in a static background.

Problems with Solutions

Problem 1. Why must gauge anomalies cancel in the Standard Model? Solution. A gauge anomaly would make the theory inconsistent: the Ward identities fail, unitarity is violated, and the theory cannot be renormalized. The SM cancels anomalies because the quark and lepton contributions add to zero: $\sum_{\text{generations}}(2N_cQ_q^2-Q_{\ell^+}^2-Q_{\ell^-}^2)=0$.

Problem 2. Write the dimension-6 operator $\mathcal O_{6\psi}=(\bar\ell\gamma^\mu\ell)(\bar e\gamma_\mu e)$ that contributes to muon decay. If the coefficient is $c/\Lambda^2$ and $c\sim1$, what limit on $\Lambda$ does the muon lifetime constraint $G_F$ imply? Solution. Since $G_F/\sqrt{2}\sim1/(2v^2)\approx1.17\times10^{-5}\,\mathrm{GeV^{-2}}$, requiring the new contribution to be smaller gives $\Lambda\gtrsim1/\sqrt{G_F}\approx300\,\mathrm{GeV}$. More precisely, precision electroweak data push this to $\Lambda\gtrsim few\,\mathrm{TeV}$.

Section summary. Anomalies and EFT show how quantum consistency and high scales shape low-energy physics.