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Part II: Quantum Chromodynamics (QCD)

Introduction to QCD

Quantum Chromodynamics (QCD) is the theory of the strong interaction, describing how quarks and gluons interact through the exchange of color charge. It is a non-Abelian gauge theory based on the $SU(3)_C$ symmetry group.

QCD exhibits two fundamental phenomena: asymptotic freedom (quarks behave as free particles at high energies) and confinement (quarks are never observed in isolation at low energies).

Color Charge and SU(3)

The Color Quantum Number

Each quark flavor comes in three "colors": red ($r$), green ($g$), and blue ($b$). Antiquarks carry anticolors: antired ($\bar{r}$), antigreen ($\bar{g}$), and antiblue ($\bar{b}$).

The three color charges form the basis of the $SU(3)_C$ gauge group, where $C$ stands for "color". This is a non-Abelian group with 8 generators, corresponding to 8 gluons.

Color Singlet Condition (Observable Hadrons):

$$ \text{Baryons: } |r\rangle|g\rangle|b\rangle - \text{totally antisymmetric} $$$$ \text{Mesons: } \frac{1}{\sqrt{3}}(|r\bar{r}\rangle + |g\bar{g}\rangle + |b\bar{b}\rangle) $$

SU(3) Generators and Structure Constants

The 8 generators of $SU(3)$ are represented by the Gell-Mann matrices $\lambda^a$ ($a = 1, 2, ..., 8$):

$$ [T^a, T^b] = if^{abc}T^c $$

where $f^{abc}$ are the structure constants of $SU(3)$. The non-zero structure constants make QCD non-Abelian, leading to gluon self-interactions.

Gluons: The Force Carriers

Eight Gluons

There are 8 gluons in QCD (compared to 1 photon in QED). Each gluon carries both a color and an anticolor:

Six Off-Diagonal Gluons:

  • $|r\bar{g}\rangle, |r\bar{b}\rangle$
  • $|g\bar{r}\rangle, |g\bar{b}\rangle$
  • $|b\bar{r}\rangle, |b\bar{g}\rangle$

Two Diagonal Gluons:

  • $\frac{1}{\sqrt{2}}(|r\bar{r}\rangle - |g\bar{g}\rangle)$
  • $\frac{1}{\sqrt{6}}(|r\bar{r}\rangle + |g\bar{g}\rangle - 2|b\bar{b}\rangle)$

The color singlet combination $\frac{1}{\sqrt{3}}(|r\bar{r}\rangle + |g\bar{g}\rangle + |b\bar{b}\rangle)$ is excluded, leaving exactly 8 physical gluons.

Gluon Self-Interaction

Unlike photons in QED, gluons carry color charge and can interact with each other. This leads to:

  • 3-gluon vertices: $ggg$ interactions
  • 4-gluon vertices: $gggg$ interactions
  • Asymptotic freedom: Anti-screening of color charge at short distances
  • Confinement: Strong force increases with distance

QCD Lagrangian:

$$ \mathcal{L}_{\text{QCD}} = \sum_{q} \bar{q}(i\gamma^\mu D_\mu - m_q)q - \frac{1}{4}G^a_{\mu\nu}G^{a\mu\nu} $$$$ D_\mu = \partial_\mu - ig_s T^a A^a_\mu, \quad G^a_{\mu\nu} = \partial_\mu A^a_\nu - \partial_\nu A^a_\mu + g_s f^{abc}A^b_\mu A^c_\nu $$

Asymptotic Freedom

Running Coupling Constant

The QCD coupling constant $\alpha_s$ depends on the energy scale $Q^2$ (or equivalently, distance scale). This "running" is described by the renormalization group equation:

$$ \alpha_s(Q^2) = \frac{\alpha_s(\mu^2)}{1 + \frac{\alpha_s(\mu^2)}{12\pi}(33 - 2n_f)\ln(Q^2/\mu^2)} $$

Leading order approximation:

$$ \alpha_s(Q^2) \approx \frac{12\pi}{(33 - 2n_f)\ln(Q^2/\Lambda^2_{\text{QCD}})} $$

where $n_f$ is the number of active quark flavors and $\Lambda_{\text{QCD}} \approx 200$ MeV is the QCD scale parameter.

High Energy (Short Distance):

$Q^2 \gg \Lambda^2_{\text{QCD}}$: $\alpha_s \to 0$

Quarks behave as free particles (perturbative QCD applies)

Low Energy (Long Distance):

$Q^2 \sim \Lambda^2_{\text{QCD}}$: $\alpha_s \sim 1$

Strong coupling regime (non-perturbative, confinement)

Nobel Prize Discovery (2004)

David Gross, Frank Wilczek, and David Politzer were awarded the Nobel Prize in Physics for discovering asymptotic freedom in QCD. The key result is that the beta function is negative:

$$ \beta(\alpha_s) = -\frac{\alpha_s^2}{2\pi}(11 - \frac{2n_f}{3}) < 0 \quad \text{for } n_f < 17 $$

The negative beta function means the coupling decreases at high energies, opposite to QED where coupling increases. This is due to gluon self-interactions creating "anti-screening".

Quark Confinement

The Confinement Phenomenon

Quarks are never observed as free particles. When you try to separate two quarks, the potential energy increases linearly with distance:

$$ V(r) \approx -\frac{4}{3}\frac{\alpha_s(r)}{r} + \kappa r $$

String tension: $\kappa \approx 1 \text{ GeV/fm} = 16 \text{ tons force}$

The linear term $\kappa r$ dominates at large distances. When enough energy is stored in the "flux tube" connecting quarks, it becomes energetically favorable to create a $q\bar{q}$ pair, resulting in hadronization rather than free quarks.

Hadronization and Fragmentation

When a quark is produced in a high-energy collision, it undergoes hadronization:

  1. High-energy quark begins moving away from collision point
  2. Color flux tube forms between quark and its color source
  3. Flux tube stretches, storing energy $E \sim \kappa r$
  4. When $E > 2m_\pi$, a $q\bar{q}$ pair is created
  5. Process repeats, creating a "jet" of hadrons

Observed Consequences:

  • All observed particles are color-neutral hadrons
  • High-energy collisions produce collimated jets of hadrons
  • Jet structure reflects underlying quark/gluon dynamics
  • No free quarks have ever been detected despite extensive searches

QCD at Colliders: Jets

Jet Production and Structure

At particle colliders like the LHC, QCD processes dominate. Quarks and gluons produced in collisions manifest as jets:

Dijet Production:

$pp \to jj + X$

Dominant process at hadron colliders

Cross section $\sim 10^8$ pb at $\sqrt{s} = 13$ TeV

Gluon Jets:

Broader and contain more particles than quark jets

Color factor $C_A = 3$ vs $C_F = 4/3$ for quarks

Jet Algorithms

Jets are reconstructed using algorithms that cluster nearby particles. Common algorithms include:

Anti-$k_t$ Algorithm (most common at LHC):

$$ d_{ij} = \min(p_{T,i}^{-2}, p_{T,j}^{-2})\frac{\Delta R_{ij}^2}{R^2} $$$$ \Delta R_{ij} = \sqrt{(\Delta\eta)^2 + (\Delta\phi)^2} $$
  • Infrared safe: Results don't change with soft emissions
  • Collinear safe: Results don't change when particles split
  • Cone-like: Produces approximately circular jets
  • Parameter $R$: Typical values $R = 0.4$ or $0.8$

Lattice QCD

Non-Perturbative Approach

At low energies where $\alpha_s \sim 1$, perturbation theory fails. Lattice QCD provides a non-perturbative approach by discretizing spacetime:

Spacetime discretization:

$$ x^\mu \to n^\mu a, \quad n^\mu \in \mathbb{Z}, \quad a = \text{lattice spacing} $$

Path integral on lattice:

$$ Z = \int \mathcal{D}U \mathcal{D}\psi \mathcal{D}\bar{\psi} \, e^{-S_G[U] - S_F[\psi, \bar{\psi}, U]} $$

Link variables $U_\mu(n) \in SU(3)$ represent gluon fields, fermion fields $\psi$ live on lattice sites. Monte Carlo methods evaluate path integral numerically.

Lattice QCD Achievements

Lattice QCD has successfully computed many non-perturbative quantities:

Hadron Masses:

  • Proton: $m_p = 938$ MeV (agreement within 2%)
  • Pion: $m_\pi = 140$ MeV
  • Entire hadron spectrum

Decay Constants:

  • Pion decay: $f_\pi = 130$ MeV
  • Kaon decay: $f_K = 156$ MeV
  • B meson decays

Structure Functions:

  • Parton distribution functions
  • Generalized parton distributions
  • Form factors

QCD Phase Diagram:

  • Quark-gluon plasma at high $T$
  • Chiral symmetry restoration
  • Critical temperature $T_c \approx 155$ MeV

Deep Inelastic Scattering (DIS)

Probing the Proton Structure

Deep inelastic scattering ($e^- + p \to e^- + X$) probes the quark and gluon structure of the proton. Key variables:

$$ Q^2 = -q^2 = -(k - k')^2 \quad \text{(virtuality of photon)} $$$$ x = \frac{Q^2}{2p \cdot q} \quad \text{(Bjorken x, momentum fraction)} $$$$ y = \frac{p \cdot q}{p \cdot k} \quad \text{(inelasticity)} $$

At high $Q^2$, the cross section exhibits Bjorken scaling: it depends on $x$ but only weakly on $Q^2$, revealing the point-like structure of quarks.

Parton Distribution Functions (PDFs)

The proton's structure is described by PDFs $f_i(x, Q^2)$, giving the probability to find parton $i$ carrying momentum fraction $x$ at scale $Q^2$:

$$ F_2(x, Q^2) = \sum_q e_q^2 x[q(x, Q^2) + \bar{q}(x, Q^2)] $$
  • Valence quarks: $uud$ for proton, carry $\sim 40\%$ of momentum
  • Sea quarks: $q\bar{q}$ pairs from gluon splitting, carry $\sim 10\%$
  • Gluons: Carry $\sim 50\%$ of momentum at $Q^2 \sim 10$ GeV$^2$

Key Takeaways

  • •QCD is based on $SU(3)_C$ color gauge symmetry with 8 gluons as force carriers
  • •Gluon self-interaction makes QCD non-Abelian, leading to asymptotic freedom and confinement
  • •Running coupling: $\alpha_s(M_Z) \approx 0.118$, increases at low energies
  • •Confinement ensures only color-neutral hadrons are observed in nature
  • •At colliders, quarks and gluons appear as jets of hadrons
  • •Lattice QCD provides non-perturbative calculations of hadron properties
  • •DIS reveals parton structure: valence quarks, sea quarks, and gluons
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