Subatomic Physics I - Lecture 5: Quarks, Gluons, and Strong Interaction, and Particle Production

Quarks in Hadrons and Classification

• Hadrons Definition: These are composite systems made entirely of quarks.
• Classification of Hadrons:     * Baryons: These possess spin $n \times 1/2$. The lightest baryons are the proton and the neutron. Quarks in baryons determine the baryon number.     * Mesons: These possess spin $n \times 1$. The lightest mesons are pions ($\pi$). They are classified as bosons.
• Stability and Occurrence:     * Most hadrons are short-lived and are not part of daily life, except for the proton and neutron.     * Baryons are produced in pairs in experimental settings.
• Quantum Numbers:     * Baryon Number ($B$): Quarks have a value of $+1/3$, while anti-quarks have $-1/3$. Consequently, baryons have $B = +1$ and anti-baryons have $B = -1$.     * Lepton Number ($L$): Debtors are assigned $+1$ for leptons and $-1$ for anti-leptons.     * Standard Model (SM) interactions conserve both $B$ and $L$ numbers.
• Prohibited Decays: The decay $p \rightarrow \pi^0 + e^+$ is energetically allowed but forbidden in the SM because it violates conservation laws:     * Baryon Number: $+1 \neq 0 + 0$.     * Lepton Number: $0 \neq 0 + (-1)$.     * The non-observation of proton decay suggests a lifetime \tau > 10^{33}\,\text{years}.
• Lightest Mesons (Pions):     * Approximate mass of $\sim 140\,MeV$.     * $\pi^+ = u\bar{d}$.     * $\pi^- = \bar{u}d$.     * $\pi^0 = \frac{1}{\sqrt{2}} { u\bar{u} - d\bar{d} }$, representing a mixture.     * Pions decay into lighter leptons, neutrinos, and photons.     * There is no specific quantum number associated with mesons, and they can be produced in any number in experiments.     * Naming of mesons vs anti-mesons is a matter of convention.

The Quark-Gluon Interaction and Quantum Chromodynamics (QCD)

• Color Charge Necessity: The introduction of color charge solves the Pauli exclusion principle violation observed in particles like the $\Delta^{++}$ baryon.     * $\Delta^{++} = uuu$ with spin-parity $J^P = \frac{3}{2}^+$.     * Configuration: $u\uparrow u\uparrow u\uparrow$.     * The total wave function (spin and exchange) appears symmetric, which is forbidden; adding the color dimension (Red, Blue, Green) makes the overall wave function antisymmetric.
• Theories of Interaction:     * Quantum Chromodynamics (QCD): The quantum field theory dealing with the strong interaction.     * Quantum Electrodynamics (QED): The equivalent theory for electromagnetic interaction.
• The Gluon:     * A massless field particle with $J^P = 1^-$, categorized as a vector boson.     * Carries simultaneous color and anti-color charges.     * Color States: Represented by a color octet and a color singlet.         * The octet includes: $r\bar{g}, r\bar{b}, g\bar{b}, g\bar{r}, b\bar{r}, b\bar{g}, \frac{1}{\sqrt{2}}(r\bar{r} - g\bar{g}), \frac{1}{\sqrt{6}}(r\bar{r} + g\bar{g} - 2b\bar{b})$.         * The singlet is $\frac{1}{\sqrt{3}}(r\bar{r} + g\bar{g} + b\bar{b})$.     * Self-Interaction: Since gluons carry color charge, they interact with themselves, a distinct property that differrentiates QCD from QED.
• Confinement and Color Neutrality:     * Hadrons are observed only as colorless (color-neutral) objects.     * Combination rules for white ($W$):         * $R + B + G = W$.         * $\bar{R} + \bar{B} + \bar{G} = W$.         * Combination of a color and its anti-color: $R + \bar{R} = W, B + \bar{B} = W, G + \bar{G} = W$.     * Actual particles like $\pi^+$ or protons are mixtures of these color states.     * Consequences: Quarks can never be observed as free particles. Attempting to separate them increases the strong interaction (unlike the Coulomb potential which decreases via $1/r^2$).

Coupling Constants and Asymptotic Freedom

• Running Coupling Constants:     * Both $\alpha_{em}$ (QED) and $\alpha_s$ (QCD) vary with $Q^2$ (energy/momentum transfer).
• QED Screening (Vacuum Polarization):     * An electron is surrounded by virtual $e^+e^-$ pairs. Positrons tend to be closer to the electron, "screening" its charge.     * The effective charge increases if the probe gets closer (high energy, large $Q^2$).     * $\alpha(m_Z) \approx \frac{1}{128}$ compared to $\alpha \approx \frac{1}{137}$ at low energies.
• QCD Antiscreening:     * While quark loops cause screening (similar to QED), gluon self-interaction loops cause "antiscreening."     * Antiscreening dominates, leading to an effective color charge that decreases as distance decreases (large $Q^2$).
• Asymptotic Freedom:     * At very high $Q^2$, $\alpha_s \rightarrow 0$. Quarks behave as almost free particles.     * Discovery credited to Wilczek, Gross, and Politzer (1973; Nobel Prize 2004).     * Formula for Running $\alpha_s$:         * $\alpha_s(Q^2) = \frac{\alpha_s(\mu^2)}{1 + \frac{\alpha_s(\mu^2)}{12\pi}(33 - 2f) \ln(\frac{Q^2}{\mu^2})}$.         * For 3 colors ($n=3$) and 6 flavors ($f=6$), the beta function is negative/positive (context dependent on sign convention for $\beta$), leading to $\alpha_s$ decreasing at short distances.
• Confinement Barrier: At low $Q^2$ (large distances $\sim 1\,fm$), $\alpha_s \approx 1$ and perturbation theory fails (the summation of diagrams diverges).

Scaling Violations of the Structure Functions

• Bjorken Scaling: The property where structure functions are independent of $Q^2$. This suggests quarks are point-like.
• Scaling Violation: Recent high-precision measurements show scaling fails at low values of $x$ (below 0.1).     * As $x$ decreases, the structure function rises with $Q^2$.     * Explanation: A virtual photon's resolution is proportional to $\sqrt{Q^2}$. At higher $Q^2$, the photon resolves more partons (sea quarks and gluons), leading to q(x, Q^2) > q(x, Q_0^2).     * Scaling violation serves as a tool to measure $\alpha_s$.     * At $Q^2 = (100\,GeV)^2$, $\alpha_s \approx 0.12$.

Particle Production in $e^+e^-$ Colliders

• Colliders Overview: $e^+e^-$ machines are excellent for clean studies of particle properties. Examples include LEP (CERN), PETRA (DESY), and SLC (Stanford).
• Center-of-Mass Energy ($\sqrt{s}$):     * Lorentz invariant variable: $s = (cP_1 + cP_2)^2$.     * For head-on collisions where $m \ll E$:         * $\sqrt{s} \approx 2E$ (Colliding beams).         * $\sqrt{s} \approx \sqrt{2mc^2E}$ (Fixed target).     * LEP (Large Electron Positron): Reached $\sqrt{s} \approx 207\,GeV$. Ended operation in 2001.     * To produce a particle of mass $M$, one needs $\sqrt{s} \ge Mc^2$. For Higgs ($M_H = 125\,GeV/c^2$) and Z ($M_Z = 91\,GeV/c^2$) production, $\sqrt{s} \ge 216\,GeV$ was ideally required.
• Fermion Production: All fermion-anti-fermion pairs can be produced. Neutrinos are only produced via weak interaction since they lack electric charge.

Lepton Pair Production and Universality

• Muon Pair Production: $e^+ e^- \rightarrow \mu^+ \mu^-$.     * Muon characteristics: $M_\mu = 105.7\,MeV/c^2$, lifetime $\approx 2 \times 10^{-6}\,s$, discovered by C. Anderson (1936).     * Primary decay: $\mu^- \rightarrow e^- + \bar{\nu}<em>e + \nu</em>\mu$.
• Tau Pair Production: $e^+ e^- \rightarrow \tau^+ \tau^-$.     * Tau characteristics: $M_\tau = 1776.82\,MeV/c^2$, lifetime $\approx 3 \times 10^{-13}\,s$, discovered at SLAC (1974-1977) by M. Perl (Nobel Prize 1995).     * Observed as a steep increase in $\sigma(e^+e^- \rightarrow e^\pm + \mu^\mp + \dots)$ at $\sqrt{s} \approx 3.55\,GeV$.     * Decay modes: $\sim 35\%$ leptonic, $\sim 65\%$ hadronic.
• Cross Section for $e^+ e^- \rightarrow \ell^+ \ell^-$:     * Differential cross section (neglecting weak interaction at low energies):         * $\frac{d\sigma}{d\Omega} = \frac{\alpha^2}{4s}(\hbar c)^2 (1 + \cos^2 \theta)$.     * Total cross section (integrated over angle):         * $\sigma = \frac{4\pi\alpha^2}{3s}(\hbar c)^2 \approx \frac{21.7}{E^2\,(\text{in } GeV^2)}\,nb$.
• Lepton Universality: $\mu$ and $\tau$ behave identically to electrons except for their mass. Since measurement matches prediction perfectly, their form factors are unity, meaning they are point-like at probed energies.

Resonance Production and the Breit-Wigner Formula

• The Breit-Wigner Formula: Describes the cross-section dependence on energy near a resonance energy $E_0$.     * $\sigma(E) = \frac{3\pi \lambda^2}{4} \frac{\Gamma_i \Gamma_f}{(E-E_0)^2 + \Gamma_{tot}^2/4}$.     * $\lambda$ is the reduced wavelength at the CMS.     * Lifetime-Width Relationship: $\tau = \frac{\hbar}{\Gamma_{tot}}$.     * Branching Ratio (BR): $BR(P \rightarrow f) = \frac{\Gamma_f}{\Gamma_{tot}}$.     * Example: Z-boson width $\Gamma_{tot}(Z) \approx 2.5\,GeV/c^2$.
• Vector Mesons ($J^P = 1^-$):     * These quark-antiquark bound states have the same quantum numbers as a virtual photon.     * $\rho(770)$: $M = 771\,MeV/c^2$, $\Gamma = 149\,MeV$, $\tau = 4.4 \times 10^{-25}\,s$.     * $\omega(782)$: $M = 781\,MeV/c^2$, $\Gamma = 8.44\,MeV$, $\tau = 7.7 \times 10^{-23}\,s$.     * $\phi(1019)$: $M = 1019\,MeV/c^2$, $\Gamma = 4.4\,MeV$, $\tau = 1.5 \times 10^{-22}\,s$.
• Strangeness ($S$):     * The $\phi$ meson decays primarily to $K^+ K^-$ or $K^0 \bar{K}^0$.     * Strange particles are produced via strong interaction (always in $s\bar{s}$ pairs) but decay only via weak interaction if they are the lightest of their type (e.g., Kaons, $M \approx 450\,MeV/c^2$).     * Strangeness Quantum Number: $S = N(\bar{s}) - N(s)$. Conserved in strong/EM, violated in weak.
• Heavy Quarks Discovery:     * J/$\psi$ (1974): Discovered simultaneously at Brookhaven (Samuel Ting) and SLAC (Burton Richter). Mass $M \approx 3097\,MeV/c^2$, indicates the charm quark ($c\bar{c}$).     * $\Upsilon$ (1977): Discovered by Leon Lederman. Mass $M = 9.46\,GeV/c^2$, indicates the beauty/bottom quark ($b\bar{b}$).     * Top Quark (1995): Discovered at FNAL. $M_{top} = 172.38\,GeV/c^2$. It is the only quark that decays before it can be confined into hadrons ($t \rightarrow Wb \approx 100\%$).

Non-resonant Hadron Production and the R-ratio

• Non-resonant Process: Kinematically allowed quark pairs are produced and سپس hadronize into jets.
• Calculation of σ(e⁺e⁻ → hadrons):     * Summing over quark species: $\sigma_f \propto z_f^2 \alpha^2$.     * Includes a factor of 3 for the three color degrees of freedom.     * $\sigma(e^+ e^- \rightarrow q_f \bar{q}_f) = 3 \times z_f^2 \times \sigma(e^+ e^- \rightarrow \mu^+ \mu^-)$.
• The R Ratio:     * $R = \frac{\sigma(e^+ e^- \rightarrow \text{hadrons})}{\sigma(e^+ e^- \rightarrow \mu^+ \mu^-)} = 3 \sum_f z_f^2$.     * R increases in steps as center-of-mass energy reaches thresholds for new quark flavors:         * For $u, d, s$: $3 \times [(\frac{2}{3})^2 + (-\frac{1}{3})^2 + (-\frac{1}{3})^2] = 3 \times \frac{6}{9} = 2$.         * Adding $c$: $3 \times \frac{10}{9} = 3.33$.         * Adding $b$: $3 \times \frac{11}{9} = 3.67$.     * If top were included, $R = \frac{15}{3} = 5$.     * The agreement of the R graph with experimental data confirms the existence of 3 colors and 5 quark flavors at those energies.

Discovery of the Gluon

• Indirect Evidence: Integrated structure functions showed quarks/anti-quarks only carry about half of the proton momentum.     * $\int_0^1 F_2^{e,N}(x)\,dx \approx 0.5$.     * This implies a particle with no electric or weak charge carries the remaining half; the gluon is the candidate.
• Direct Evidence (1979): Discovered at PETRA (DESY) in $e^+e^-$ collisions at $\sqrt{s} \approx 30\,GeV$ as radiation from quarks.     * Evidence came in the form of "three-jet events" ($e^+e^- \rightarrow q\bar{q}g$) where a third jet arises from a gluon.     * The angular distribution of these hadrons confirmed the gluon as a spin 1 boson.