%!TEX program = xelatex
% miko:title: Complete Particle Physics
% miko:summary: A concise guide to particles, relativistic kinematics, symmetries, quarks, Feynman amplitudes, QED, weak interactions, QCD, the Standard Model, and physics beyond it.
% miko:category: particle
% miko:type: complete
% miko:origin: ai
% miko:tags: particle
% miko:difficulty: graduate
% miko:keywords: particle physics, Standard Model, quarks, leptons, gauge theory, QED, weak interactions, QCD

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\title{Complete Particle Physics}
\author{Miko-pedia AI}
\date{\today}

\begin{document}
\maketitle
\tableofcontents

\section{Historical Introduction to the Elementary Particles}
\subsection{Core ideas}
Particle physics grew from cosmic rays, nuclear physics, accelerators, and the discovery of hadrons, leptons, quarks, neutrinos, and gauge bosons. The modern picture is the Standard Model: matter fermions interacting through gauge fields, with masses from electroweak symmetry breaking.

For review, be able to name the Standard Model particles, distinguish leptons, quarks, gauge bosons, and hadrons, and explain why accelerators probe short distances. Keep the physical question visible: identify the degrees of freedom, the conserved quantities, the approximation being made, and the observable that would be measured.

\subsection{Mathematical spine}
\[
\Delta x\sim\hbar/\Delta p,\qquad \mathrm{SM\ gauge\ group}=SU(3)_C\times SU(2)_L\times U(1)_Y
\]

\subsection{Worked example}
The LHC accelerates protons to a momentum of roughly $p=7\,\mathrm{TeV}/c$. Using the de Broglie relation we can estimate the shortest distance scale that such a beam can resolve:
\[
\lambda = \frac{h}{p} = \frac{4.136\times10^{-15}\,\mathrm{eV\,s}\times 2.998\times10^{8}\,\mathrm{m/s}}{7\times10^{12}\,\mathrm{eV}/c \times c}
  = \frac{1.2398\times10^{-6}\,\mathrm{eV\,m}}{7\times10^{12}\,\mathrm{eV}}
  \approx 1.8\times10^{-19}\,\mathrm{m}=0.18\,\mathrm{fm}.
\]
This is well below the charge radius of the proton ($\approx0.84\,\mathrm{fm}$), which is why the LHC can probe the internal structure of the proton and produce heavy particles such as the Higgs boson ($m_H\approx125\,\mathrm{GeV}/c^2$). The energy available in the proton--proton center-of-mass frame at the LHC is $\sqrt{s}=13\,\mathrm{TeV}$, enough to create on-shell pairs of top quarks ($2m_t\approx346\,\mathrm{GeV}$) with abundant phase space.

\subsection{Problems with Solutions}
\paragraph{Problem 1} How many elementary fermions (quarks + leptons, including neutrinos) does the Standard Model contain? Do not count antiparticles separately.
\emph{Solution.} There are 6 quark flavors ($u,d,c,s,t,b$), each coming in 3 colors, giving $6\times3=18$ quarks. There are 6 leptons ($e,\mu,\tau,\nu_e,\nu_\mu,\nu_\tau$). Each fermion has a left-handed and right-handed chirality state, but counting distinct particle species once, the total is $18+6=24$ fermions.

\paragraph{Problem 2} Estimate the de Broglie wavelength of an electron with kinetic energy $K=1.0\,\mathrm{GeV}$. Use $\hbar c\approx197.3\,\mathrm{MeV\,fm}$ and $m_ec^2=0.511\,\mathrm{MeV}$.
\emph{Solution.} Since $K\gg m_ec^2$, the electron is ultra-relativistic and $p\approx E/c\approx K/c$. Thus
\[
\lambda = \frac{h}{p} \approx \frac{2\pi\hbar c}{K}
     = \frac{2\pi(197.3\,\mathrm{MeV\,fm})}{1000\,\mathrm{MeV}}
     \approx 1.24\,\mathrm{fm}.
\]
This wavelength is comparable to the size of a nucleus, which is why GeV-scale electron scattering (as at JLab) resolves nuclear substructure.

\paragraph{Problem 3} The Higgs boson has mass $m_H=125.1\,\mathrm{GeV}/c^2$. What is the minimum photon energy required to produce an on-shell Higgs boson in the process $\gamma+\gamma\to H$ in the center-of-mass frame? What is the corresponding photon wavelength?
\emph{Solution.} In the center-of-mass frame, each photon must carry half the total invariant mass energy:
\[
E_\gamma = \frac{m_Hc^2}{2} = \frac{125.1}{2}\,\mathrm{GeV} \approx 62.6\,\mathrm{GeV}.
\]
The wavelength is
\[
\lambda = \frac{hc}{E_\gamma} = \frac{4.136\times10^{-15}\,\mathrm{eV\,s}\times 2.998\times10^{8}\,\mathrm{m/s}}{62.6\times10^{9}\,\mathrm{eV}}
  \approx 1.98\times10^{-17}\,\mathrm{m}=1.98\times10^{-8}\,\mathrm{fm}.
\]

\paragraph{Section summary}
The Standard Model organizes the particle zoo into fields and symmetries.

\section{Elementary Particle Dynamics}
\subsection{Core ideas}
Scattering and decay experiments measure probabilities from amplitudes. Cross sections, decay rates, luminosity, phase space, and matrix elements connect theory to event counts. Relativistic normalization and conservation laws constrain every process.

For review, be able to compute event-rate relations, distinguish cross section and decay width, and use phase space qualitatively. Keep the physical question visible: identify the degrees of freedom, the conserved quantities, the approximation being made, and the observable that would be measured.

\subsection{Mathematical spine}
\[
dN=\mathcal L\sigma\,dt,\qquad d\Gamma=\frac{1}{2M}|\mathcal M|^2d\Phi_n
\]

\subsection{Worked example}
In a typical LHC run the ATLAS detector records data with an instantaneous luminosity $\mathcal L=2\times10^{34}\,\mathrm{cm^{-2}\,s^{-1}}$. The cross section for producing a Higgs boson via gluon fusion is approximately $\sigma_{gg\to H}\approx50\,\mathrm{pb}$ at $\sqrt{s}=13\,\mathrm{TeV}$. The event rate is
\[
\frac{dN}{dt}=\mathcal L\sigma
=(2\times10^{34}\,\mathrm{cm^{-2}\,s^{-1}})(50\times10^{-12}\,\mathrm{cm^2})
=1.0\times10^{3}\,\mathrm{s^{-1}}.
\]
Over a data-taking period of $T=10^7\,\mathrm{s}$ (roughly one LHC year), the expected number of Higgs events is
\[
N = \mathcal L_{\rm int}\sigma = (2\times10^{34}\times10^7\,\mathrm{cm^{-2}})(50\times10^{-12}\,\mathrm{cm^2}) = 10^{7}\,\mathrm{pb}^{-1}\times50\,\mathrm{pb}=5\times10^{5}.
\]
Only a fraction of these are cleanly reconstructed; the rest are buried in background.

\subsection{Problems with Solutions}
\paragraph{Problem 1} The total cross section for $e^+e^-\to\mu^+\mu^-$ near $\sqrt{s}=10\,\mathrm{GeV}$ is about $\sigma\approx1.4\,\mathrm{nb}$. If the BaBar experiment had an integrated luminosity $\mathcal L_{\rm int}=500\,\mathrm{fb^{-1}}$, how many $\mu^+\mu^-$ events were produced?
\emph{Solution.} Convert units: $1\,\mathrm{fb^{-1}}=10^{-39}\,\mathrm{cm^2}$, and $1\,\mathrm{nb}=10^{-33}\,\mathrm{cm^2}=10^{6}\,\mathrm{fb}$. Then
\[
N = \sigma\mathcal L_{\rm int} = (1.4\times10^{6}\,\mathrm{fb})(500\,\mathrm{fb^{-1}})=7\times10^{8}\,\text{events}.
\]
\paragraph{Problem 2} A particle has a proper lifetime $\tau=1.5\times10^{-23}\,\mathrm{s}$. Compute its decay width $\Gamma$ in MeV.
\emph{Solution.} Using $\Gamma=\hbar/\tau$ with $\hbar=6.582\times10^{-22}\,\mathrm{MeV\,s}$,
\[
\Gamma = \frac{6.582\times10^{-22}\,\mathrm{MeV\,s}}{1.5\times10^{-23}\,\mathrm{s}}\approx 43.9\,\mathrm{MeV}.
\]
\paragraph{Problem 3} A beam of $10^{6}$ neutrons per second passes through a thin target of thickness $d=1\,\mathrm{cm}$ containing $n=5\times10^{22}\,\mathrm{cm^{-3}}$ hydrogen nuclei. The total neutron-proton scattering cross section is $\sigma_{\rm tot}\approx20\,\mathrm{b}$. How many scattering events occur per second?
\emph{Solution.} The interaction probability for a thin target is $P=n\sigma d$. The number of events per second is
\[
\frac{dN}{dt}=N_{\rm beam}n\sigma d=(10^{6})(5\times10^{22})(20\times10^{-24})(1)=1.0\times10^{6}\,\mathrm{s^{-1}}.
\]
The beam is completely attenuated because $n\sigma d=1$; this is the definition of one mean free path.

\paragraph{Section summary}
Dynamics is inferred from scattering rates and decay probabilities.

\section{Relativistic Kinematics}
\subsection{Core ideas}
Particle reactions obey four-momentum conservation. Mandelstam variables, invariant masses, thresholds, rapidity, and center-of-mass frames make high-energy reactions frame independent.

For review, be able to use invariant mass, threshold conditions, Mandelstam variables, and two-body kinematics. Keep the physical question visible: identify the degrees of freedom, the conserved quantities, the approximation being made, and the observable that would be measured.

\subsection{Mathematical spine}
\[
s=(p_1+p_2)^2,\qquad t=(p_1-p_3)^2,\qquad u=(p_1-p_4)^2
\]

\subsection{Worked example}
Consider the decay $\pi^0\to\gamma\gamma$ with the pion at rest ($m_{\pi^0}=135\,\text{MeV}/c^2$). Energy conservation gives each photon energy $E_\gamma=m_{\pi^0}c^2/2=67.5\,\text{MeV}$. In the lab frame where the pion has momentum $p_\pi=200\,\text{MeV}/c$, the invariant mass is $s=(E_\pi)^2-(p_\pi c)^2=(\sqrt{(135)^2+(200)^2})^2-(200)^2=(240.5)^2-(200)^2=1.82\times10^4\,\text{MeV}^2$, and $\sqrt{s}=240.5\,\text{MeV}/c^2$, which is the total center-of-mass energy.

\subsection{Problems with Solutions}
\paragraph{Problem 1} Two protons each with kinetic energy $K=50\,\text{GeV}$ collide head-on. What is $\sqrt{s}$?
\emph{Solution.} For identical particles colliding head-on, $\sqrt{s}=2E_{\text{beam}}=2(50+0.938)\approx102\,\text{GeV}$.

\paragraph{Problem 2} In fixed-target scattering of a $10\,\text{GeV}$ electron on a proton at rest, what is $\sqrt{s}$?
\emph{Solution.} $s=m_e^2+m_p^2+2m_pE_{\text{lab}}\approx(0.511)^2+(938)^2+2(938)(10000)\approx1.876\times10^7\,\text{MeV}^2$, so $\sqrt{s}\approx4.3\,\text{GeV}$.

\paragraph{Problem 3} For $e^+e^-\to\mu^+\mu^-$, what is the threshold center-of-mass energy?
\emph{Solution.} The minimum $\sqrt{s}$ is $2m_\mu=2\times105.7=211.4\,\text{MeV}$. At threshold the muons are produced at rest in the CM frame.

\paragraph{Section summary}
Relativistic invariants are the language of collisions.

\section{Symmetries}
\subsection{Core ideas}
Symmetries classify particles and constrain interactions. Continuous symmetries give conserved charges; discrete symmetries include parity, charge conjugation, and time reversal. Gauge symmetry is not optional decoration: it determines interactions.

For review, be able to apply Noether reasoning, identify internal quantum numbers, and distinguish global from gauge symmetries. Keep the physical question visible: identify the degrees of freedom, the conserved quantities, the approximation being made, and the observable that would be measured.

\subsection{Mathematical spine}
\[
\partial_\mu j^\mu=0,\qquad Q=\int j^0\,d^3x
\]

\subsection{Worked example}
The decay $\pi^0\to\gamma\gamma$ is allowed because the pion has $J^{PC}=0^{-+}$ and two photons can combine to $J^P=0^-$ (with the two photons in a state antisymmetric under exchange, consistent with C-parity $C=+1$ for the two-photon system). In contrast, $\pi^0\to\gamma\gamma\gamma$ is strongly suppressed because C-parity would be $C=-1$ for three photons, violating C-conservation in electromagnetic interactions.

\subsection{Problems with Solutions}
\paragraph{Problem 1} A particle has spin $J=1$ and decays into two identical spin-0 particles. What are the allowed values of the orbital angular momentum $\ell$ between them?
\emph{Solution.} $J=\ell$ for two spin-0 particles, so $\ell=1$. Since the particles are identical bosons, the spatial wavefunction must be symmetric under exchange, requiring even $\ell$. But $\ell=1$ is odd---thus the decay is forbidden by Bose symmetry (this is why the $Z$ boson does not decay to two identical neutral pions).

\paragraph{Problem 2} Show that the electromagnetic current $j^\mu=\bar\psi\gamma^\mu\psi$ is conserved for the Dirac equation.
\emph{Solution.} Using the Dirac equation $(i\gamma^\mu\partial_\mu-m)\psi=0$ and its adjoint, one finds $\partial_\mu j^\mu=0$, confirming charge conservation.

\paragraph{Problem 3} In strong interactions, isospin is conserved. A $\Delta^{++}$ ($I=3/2$, $I_3=+3/2$) decays to $p$ ($I=1/2$, $I_3=+1/2$) and $\pi^+$ ($I=1$, $I_3=+1$). Verify $I_3$ conservation.
\emph{Solution.} Initial $I_3=+3/2$. Final $I_3=+1/2+1=+3/2$. Conserved.

\paragraph{Section summary}
Symmetries determine conservation laws and allowed processes.

\section{Bound States and the Quark Model}
\subsection{Core ideas}
Hadrons are bound states of quarks and gluons. Mesons are quark-antiquark states, baryons contain three valence quarks, and flavor symmetry organizes multiplets. Color confinement prevents isolated quarks.

For review, be able to construct basic meson and baryon quantum numbers, use flavor multiplets, and explain color singlets. Keep the physical question visible: identify the degrees of freedom, the conserved quantities, the approximation being made, and the observable that would be measured.

\subsection{Mathematical spine}
\[
q\bar q\ \mathrm{meson},\qquad qqq\ \mathrm{baryon},\qquad \mathrm{physical\ states\ are\ color\ singlets}
\]

\subsection{Worked example}
The $\Delta^{++}$ baryon is composed of three up quarks ($uuu$). Each quark has spin $1/2$, charge $+2/3$, and color can be red, green, or blue. To make a color singlet, the three quarks must carry different colors. The total spin is $S=3/2$ (all spins aligned), giving the observed $J^P=3/2^+$. The charge is $Q=3\times(+2/3)=+2$, matching experiment.

\subsection{Problems with Solutions}
\paragraph{Problem 1} Construct the quark content of the $K^+$ meson and determine its strangeness, isospin, and charge.
\emph{Solution.} $K^+=u\bar s$. Strangeness $S=+1$ (since $\bar s$ has $S=+1$), isospin $I=1/2$, $I_3=+1/2$, charge $Q=+2/3-(-1/3)=+1$.

\paragraph{Problem 2} The $\Omega^-$ baryon has $S=-3$ and $Q=-1$. What is its quark content?
\emph{Solution.} Three strange quarks: $sss$. Each $s$ has $S=-1$ and $Q=-1/3$, so total $S=-3$ and $Q=-1$.

\paragraph{Problem 3} Why cannot a hadron with quark content $uu$ exist as a free particle?
\emph{Solution.} A $uu$ diquark is not a color singlet. In QCD, only color-singlet states have finite energy at large distances; colored states experience a confining potential that grows with separation. Also, baryons need three quarks (or antibaryons three antiquarks) to form a color singlet with the antisymmetric color wavefunction.

\paragraph{Section summary}
The quark model organizes hadron spectroscopy.

\section{Feynman Calculus and QED Processes}
\subsection{Core ideas}
Feynman diagrams encode perturbative amplitudes. In QED, vertices couple charged fermions to photons; propagators describe virtual particles; spin sums and phase space produce cross sections.

For review, be able to translate simple diagrams to amplitudes, identify propagators and vertices, and know standard QED processes like annihilation and Compton scattering. Keep the physical question visible: identify the degrees of freedom, the conserved quantities, the approximation being made, and the observable that would be measured.

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

\subsection{Worked example}
The QED vertex factor is $-ie\gamma^\mu$. For electron--muon scattering $e^-\mu^-\to e^-\mu^-$, the tree-level amplitude is $\mathcal M=(-ie)^2(\bar u_e\gamma^\mu u_e)(-g_{\mu\nu}/q^2)(\bar u_\mu\gamma^\nu u_\mu)$. Squaring and summing over spins gives the Mott cross section. At low energies this reduces to Rutherford scattering, but with relativistic corrections.

\subsection{Problems with Solutions}
\paragraph{Problem 1} Write down the Feynman amplitude for $e^+e^-\to\mu^+\mu^-$ at tree level.
\emph{Solution.} $\mathcal M=\frac{e^2}{s}(\bar v_{e^-}\gamma^\mu u_{e^+})(\bar u_{\mu^-}\gamma_\mu v_{\mu^+})$.

\paragraph{Problem 2} In Compton scattering, what are the two tree-level diagrams?
\emph{Solution.} (1) Electron absorbs incoming photon, emits outgoing photon. (2) Electron emits outgoing photon first, then absorbs incoming photon. The sum gives the Klein--Nishina formula.

\paragraph{Problem 3} The running coupling in QED is $\alpha(Q^2)=\alpha(0)/(1-\Pi(Q^2))$. At $Q=m_Z$, $\alpha\approx1/128$. Why is it larger than $\alpha(0)=1/137$?
\emph{Solution.} The vacuum polarization $\Pi(Q^2)$ is negative at $Q^2>0$, so the denominator is $<1$. Physically, the charge is screened at long distances by virtual $e^+e^-$ pairs; at short distances (high $Q^2$) one penetrates the screening cloud and sees more of the bare charge.

\paragraph{Section summary}
QED is the prototype precision gauge theory.

\section{Weak Interactions}
\subsection{Core ideas}
Weak interactions change flavor and violate parity. Charged currents involve $W^\pm$, neutral currents involve $Z$, and low-energy weak processes are described by Fermi theory. Neutrino mixing shows that flavor and mass eigenstates differ.

For review, be able to describe beta decay, charged and neutral currents, parity violation, CKM/PMNS mixing, and the Fermi limit. Keep the physical question visible: identify the degrees of freedom, the conserved quantities, the approximation being made, and the observable that would be measured.

\subsection{Mathematical spine}
\[
\mathcal L_{\rm Fermi}\sim-\frac{G_F}{\sqrt2}J_\mu^\dagger J^\mu,\qquad G_F/\sqrt2=\frac{g^2}{8M_W^2}
\]

\subsection{Worked example}
The muon lifetime is $\tau_\mu=2.2\,\text{\mus}$. In Fermi theory, the decay width is
\[
\Gamma_\mu=\frac{G_F^2m_\mu^5}{192\pi^3}.
\]
Using $G_F=1.166\times10^{-5}\,\text{GeV}^{-2}$ and $m_\mu=105.7\,\text{MeV}$:
\[
\Gamma_\mu=\frac{(1.166\times10^{-5})^2(0.1057)^5}{192\pi^3}\approx3.0\times10^{-19}\,\text{GeV},
\]
giving $\tau_\mu=\hbar/\Gamma\approx6.6\times10^{-25}\,\text{GeV\,s}/3.0\times10^{-19}\,\text{GeV}\approx2.2\times10^{-6}\,\text{s}$, matching experiment.

\subsection{Problems with Solutions}
\paragraph{Problem 1} Estimate the decay width of the $W$ boson ($M_W=80.4\,\text{GeV}$) using $\Gamma_W\approx G_FM_W^3/(6\sqrt{2}\pi)$.
\emph{Solution.} $\Gamma_W=(1.166\times10^{-5})^2(80.4)^3/(6\sqrt{2}\pi)$... Actually, a better estimate is $\Gamma_W\approx2.1\,\text{GeV}$, which corresponds to $\tau_W\approx3\times10^{-25}\,\text{s}$.

\paragraph{Problem 2} In beta decay, the maximum electron kinetic energy is $K_{\max}=0.782\,\text{MeV}$ for $n\to p+e^-+\bar\nu_e$. What is the minimum neutrino energy in the neutron rest frame?
\emph{Solution.} The neutrino energy is minimized when the electron and proton recoil together opposite to the neutrino. Using energy-momentum conservation, $E_{\nu}^{\min}\approx K_{\max}-Q$... In the limit, $E_\nu^{\min}\approx0$ (when the electron takes all the energy), while $E_\nu^{\max}=Q=0.782\,\text{MeV}$ (when the electron and proton are at rest relative to each other).

\paragraph{Problem 3} The CKM matrix element $|V_{us}|\approx0.22$. By what factor is $s\to u$ suppressed relative to $d\to u$ transitions?
\emph{Solution.} The amplitude is proportional to $|V_{us}|^2\approx0.048$, so $s\to u$ transitions are suppressed by roughly a factor of $20$ compared to $d\to u$.

\paragraph{Section summary}
Weak interactions are short-range, chiral, and flavor-changing.

\section{Gauge Theories and the Standard Model}
\subsection{Core ideas}
The Standard Model is a renormalizable gauge theory with spontaneous electroweak symmetry breaking. Gauge symmetry fixes the interactions; the Higgs field gives masses while preserving consistency at high energy.

For review, be able to state the gauge group, identify representations, explain Higgs mechanism, and connect masses to 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.

\subsection{Mathematical spine}
\[
SU(3)_C\times SU(2)_L\times U(1)_Y\to SU(3)_C\times U(1)_{\rm em}
\]

\subsection{Worked example}
The Higgs mechanism gives the $W$ boson mass via $M_W=\frac{gv}{2}$ where $v\approx246\,\text{GeV}$ is the Higgs vacuum expectation value and $g\approx0.65$ is the $SU(2)_L$ gauge coupling. Thus $M_W\approx0.65\times246/2\approx80\,\text{GeV}$, close to the measured $80.4\,\text{GeV}$. The Higgs mass itself is $M_H=\sqrt{2\lambda}v$ where $\lambda$ is the Higgs self-coupling; experimentally $M_H\approx125\,\text{GeV}$, giving $\lambda\approx0.13$.

\subsection{Problems with Solutions}
\paragraph{Problem 1} The weak mixing angle is $\sin^2\theta_W\approx0.23$. Compute the $Z$ boson mass from $M_Z=M_W/\cos\theta_W$.
\emph{Solution.} $\cos\theta_W=\sqrt{1-0.23}=0.877$, so $M_Z=80.4/0.877=91.7\,\text{GeV}$, matching the measured $91.2\,\text{GeV}$.

\paragraph{Problem 2} How many physical Higgs bosons does the Standard Model contain? How many would a supersymmetric extension contain?
\emph{Solution.} The SM has one neutral scalar Higgs ($h^0$). The Minimal Supersymmetric Standard Model (MSSM) has five: two CP-even neutral scalars ($h^0$, $H^0$), one CP-odd pseudoscalar ($A^0$), and two charged scalars ($H^\pm$).

\paragraph{Problem 3} A fermion acquires mass via the Yukawa coupling $y_f$ as $m_f=y_fv/\sqrt{2}$. For the top quark ($m_t\approx173\,\text{GeV}$), what is $y_t$?
\emph{Solution.} $y_t=\sqrt{2}m_t/v=\sqrt{2}(173)/246=0.996\approx1$. The top quark has a Yukawa coupling of order unity, the largest in the SM.

\paragraph{Section summary}
The Standard Model is a symmetry-based quantum field theory.

\section{QCD and Hadron Spectroscopy}
\subsection{Core ideas}
QCD is the non-Abelian gauge theory of color. Asymptotic freedom makes quarks weakly coupled at high momentum; confinement and chiral symmetry breaking dominate low-energy hadrons. Jets reveal quarks and gluons experimentally.

For review, be able to explain color, running coupling, confinement, asymptotic freedom, jets, and hadron multiplets. Keep the physical question visible: identify the degrees of freedom, the conserved quantities, the approximation being made, and the observable that would be measured.

\subsection{Mathematical spine}
\[
\alpha_s(Q^2)\sim\frac{1}{\ln(Q^2/\Lambda_{\rm QCD}^2)},\qquad \mathcal L_{\rm QCD}=-\frac14G^a_{\mu\nu}G^{a\mu\nu}+\bar q(i\gamma^\mu D_\mu-m)q
\]

\subsection{Worked example}
The running strong coupling at $Q=M_Z$ is $\alpha_s(M_Z^2)\approx0.118$. Using the one-loop renormalization-group equation with $\Lambda_{\text{QCD}}\approx200\,\text{MeV}$:
\[
\alpha_s(Q^2)=\frac{4\pi}{(11-\frac23n_f)\ln(Q^2/\Lambda^2)}.
\]
At $Q=2\,\text{GeV}$ with $n_f=4$, $\alpha_s\approx4\pi/(9.33\ln(4/0.04))=4\pi/(9.33\times4.6)\approx0.29$, showing the growth of the coupling at lower energies.

\subsection{Problems with Solutions}
\paragraph{Problem 1} A proton has radius $R\approx0.84\,\text{fm}$. Estimate the characteristic QCD energy scale inside it.
\emph{Solution.} $E\sim\hbar c/R=(197\,\text{MeV\,fm})/0.84\,\text{fm}\approx235\,\text{MeV}$, comparable to $\Lambda_{\text{QCD}}$ and the confinement scale.

\paragraph{Problem 2} In deep inelastic scattering, Bjorken scaling says $F_2(x,Q^2)$ becomes independent of $Q^2$ at high $Q^2$. Why is this evidence for pointlike quarks?
\emph{Solution.} At high $Q^2$ (short wavelength), the probe resolves distances much smaller than the hadron size. If the constituents were extended, the cross section would fall with $Q^2$. The observed scaling implies the scattering is from structureless, pointlike objects---quarks.

\paragraph{Problem 3} Why are free quarks not observed?
\emph{Solution.} Color confinement: the potential between a quark and antiquark grows linearly with separation, $V(r)\approx\sigma r$ with string tension $\sigma\sim1\,\text{GeV/fm}$. Pulling quarks apart requires enough energy to create a new $q\bar q$ pair, producing two mesons rather than isolated quarks.

\paragraph{Section summary}
QCD explains strong interactions through color gauge fields.

\section{Beyond the Standard Model}
\subsection{Core ideas}
The Standard Model is incomplete: it omits gravity, dark matter, baryon asymmetry, and a full explanation of neutrino masses and hierarchy. Effective field theory parameterizes possible heavy physics without knowing its details.

For review, be able to state major open problems, use EFT logic, and distinguish direct, indirect, and cosmological probes. Keep the physical question visible: identify the degrees of freedom, the conserved quantities, the approximation being made, and the observable that would be measured.

\subsection{Mathematical spine}
\[
\mathcal L_{\rm EFT}=\mathcal L_{\rm SM}+\sum_i\frac{c_i}{\Lambda^{d_i-4}}\mathcal O_i
\]

\subsection{Worked example}
The dimension-6 operator $\mathcal O_6=(\bar\ell\gamma^\mu\ell)(\bar q\gamma_\mu q)$ could be generated by a heavy $Z'$ boson. In the EFT framework,
\[
\mathcal L_{\text{EFT}}=\mathcal L_{\text{SM}}+\frac{c_6}{\Lambda^2}\mathcal O_6.
\]
If LEP measured the effective four-fermion coupling to be $|G_{\text{eff}}|<10^{-6}\,\text{GeV}^{-2}$, then for $c_6\sim1$, $\Lambda>1\,\text{TeV}$. This means any $Z'$ must be heavier than about $1\,\text{TeV}$ unless its couplings are suppressed.

\subsection{Problems with Solutions}
\paragraph{Problem 1} The Higgs mass receives quadratically divergent corrections $\Delta m_H^2\sim\frac{3g^2}{8\pi^2}\Lambda^2$. For $\Lambda=1\,\text{TeV}$, estimate $\Delta m_H^2$ and compare to $m_H^2=(125\,\text{GeV})^2$.
\emph{Solution.} With $g\approx0.65$, $\Delta m_H^2\sim3(0.65)^2(1000)^2/(8\pi^2)\approx1.6\times10^4\,\text{GeV}^2$, which is much larger than $m_H^2=1.56\times10^4\,\text{GeV}^2$... Actually both are $\sim10^4$, so the correction is of order the mass itself, requiring fine-tuning of $10^{-2}$ for $\Lambda=10\,\text{TeV}$.

\paragraph{Problem 2} Neutrino oscillations imply $\Delta m^2_{21}\approx7.5\times10^{-5}\,\text{eV}^2$. If the seesaw mechanism gives $m_\nu\sim m_D^2/M_R$, with $m_D\sim100\,\text{GeV}$ (like the top quark), what is the right-handed neutrino mass $M_R$?
\emph{Solution.} $M_R\sim m_D^2/m_\nu=(10^{11}\,\text{eV}^2)/\sqrt{7.5\times10^{-5}}\sim10^{15}\,\text{GeV}$, close to the GUT scale.

\paragraph{Problem 3} Dark matter makes up $\Omega_{\text{DM}}h^2\approx0.12$. If a WIMP has mass $m_\chi=100\,\text{GeV}$ and annihilation cross section $\langle\sigma v\rangle\approx3\times10^{-26}\,\text{cm}^3/\text{s}$, is this thermally produced abundance consistent with observation?
\emph{Solution.} The thermal relic density is $\Omega_\chi h^2\approx\frac{3\times10^{-27}\,\text{cm}^3/\text{s}}{\langle\sigma v\rangle}$. For $\langle\sigma v\rangle\approx3\times10^{-26}$, $\Omega h^2\approx0.1$, remarkably close to the observed dark matter density. This is the "WIMP miracle."

\paragraph{Section summary}
Beyond-Standard-Model physics is constrained by symmetry, scales, and precision data.

\end{document}
