10.4 Quark model and the CKM matrix

The quark model and CKM matrix are crucial components of the Standard Model. They explain how quarks combine to form hadrons and how they mix through weak interactions. This framework helps us understand particle physics phenomena like CP violation and flavor-changing decays.

The quark model classifies quarks into six flavors with specific properties, while the CKM matrix describes quark mixing in weak interactions. Together, they provide a powerful framework for understanding fundamental particle interactions and CP violation in the universe.

Quark Classification and Properties

Quark Flavors and Generations

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• Quarks are elementary particles that serve as the fundamental building blocks of hadrons (particles that experience the strong nuclear force)
• There are six types (flavors) of quarks: up (u), down (d), charm (c), strange (s), top (t), and bottom (b)
  • Quarks are grouped into three generations: (u, d), (c, s), and (t, b)
  • Each generation consists of an up-type quark (electric charge +2/3) and a down-type quark (electric charge -1/3)
  • The masses of the quarks increase with each generation, with the u and d being the lightest and the t and b being the heaviest

Quark Charges and Quantum Numbers

• Quarks have fractional electric charges: +2/3 for u, c, and t; -1/3 for d, s, and b
  • The fractional charges of quarks combine to form the integer charges of hadrons (protons, neutrons)
• Quarks carry color charge (red, green, or blue), which is the source of the strong interaction
  • Color charge is a quantum number that governs the strong interaction between quarks, analogous to electric charge in electromagnetism
  • Quarks can change their color charge through the exchange of gluons, the carriers of the strong force
• Quarks have intrinsic spin of 1/2, making them fermions
  • As fermions, quarks obey the Pauli exclusion principle, which states that no two identical fermions can occupy the same quantum state simultaneously
  • Quarks follow Fermi-Dirac statistics, which describes the statistical behavior of particles with half-integer spin

Quark Confinement and Hadron Formation

• Quarks are never observed in isolation due to color confinement
  • Color confinement is the phenomenon where quarks cannot be separated from each other at low energies
  • The strong force between quarks increases with distance, preventing them from existing as free particles
• Quarks form color-neutral hadrons: baryons (three quarks) and mesons (quark-antiquark pairs)
  • Baryons, such as protons (uud) and neutrons (udd), consist of three quarks with different color charges (red, green, blue) that combine to form a color-neutral state
  • Mesons, such as pions ($\pi^+$ (u$\bar{d}$), $\pi^-$ ($\bar{u}$d)) and kaons ($K^+$ (u$\bar{s}$), $K^-$ ($\bar{u}$s)), are composed of a quark and an antiquark with opposite color charges that cancel out
• The masses of the quarks span a wide range, with the top quark being the heaviest (173 GeV/$c^2$) and the up and down quarks being the lightest (2-8 MeV/$c^2$)
  • The large mass differences between generations have important implications for particle physics, such as the stability of hadrons and the decay patterns of heavy quarks

Quark Mixing and the CKM Matrix

Quark Mixing and Weak Interaction Eigenstates

• Quark mixing refers to the phenomenon where the weak interaction eigenstates of quarks are different from their mass eigenstates
  • Mass eigenstates (d, s, b) are the states with definite masses that propagate through space-time
  • Weak interaction eigenstates (d', s', b') are the states that participate in weak interactions, such as beta decay
• The mixing of quark flavors is described by the Cabibbo-Kobayashi-Maskawa (CKM) matrix, a 3x3 unitary matrix that relates the mass eigenstates to the weak interaction eigenstates
  • The CKM matrix elements, denoted as $V_{ij}$, represent the probability amplitude of a transition from a quark of flavor $i$ to a quark of flavor $j$ through the weak interaction
  • The unitarity of the CKM matrix ensures the conservation of probability in quark transitions

Structure and Parameterization of the CKM Matrix

• The CKM matrix is nearly diagonal, indicating that mixing between generations is suppressed
  • The diagonal elements ($V_{ud}$, $V_{cs}$, $V_{tb}$) are close to 1, while the off-diagonal elements are small
  • The suppression of inter-generational mixing is known as the "hierarchy" of the CKM matrix
• The off-diagonal elements of the CKM matrix are small but non-zero, allowing for flavor-changing weak decays
  • Flavor-changing weak decays, such as $s \to u + W^-$ or $b \to c + W^-$, are crucial for understanding CP violation and the matter-antimatter asymmetry in the universe
  • The magnitudes of the off-diagonal elements determine the branching ratios and decay rates of flavor-changing processes
• The CKM matrix is parameterized by three mixing angles ($\theta_{12}$, $\theta_{23}$, $\theta_{13}$) and one complex phase ($\delta$)
  • The mixing angles describe the rotation between the mass eigenstates and the weak interaction eigenstates in three-dimensional flavor space
  • The complex phase $\delta$ is responsible for CP violation in the quark sector, as it introduces an irreducible complex phase in the CKM matrix

CKM Matrix Implications for CP Violation

CP Violation and the Kobayashi-Maskawa Mechanism

• CP violation refers to the violation of the combined symmetry of charge conjugation (C) and parity (P)
  • C symmetry interchanges particles and antiparticles, while P symmetry inverts spatial coordinates
  • CP violation is a necessary condition for explaining the observed matter-antimatter asymmetry in the universe
• The complex phase $\delta$ in the CKM matrix introduces an irreducible CP-violating phase, leading to CP violation in the weak interactions of quarks
  • This mechanism, known as the Kobayashi-Maskawa mechanism, was proposed in 1973 by Makoto Kobayashi and Toshihide Maskawa
  • The Kobayashi-Maskawa mechanism requires at least three generations of quarks to accommodate a non-trivial complex phase in the CKM matrix
• CP violation manifests in the decays of certain mesons, such as neutral kaons ($K^0$) and B mesons ($B^0$)
  • The observed CP violation in these systems, such as the difference in decay rates between $K^0 \to \pi^+\pi^-$ and $\bar{K}^0 \to \pi^+\pi^-$, provides strong evidence for the CKM mechanism
  • The measured CP-violating parameters, such as $\epsilon_K$ for kaons and $\sin(2\beta)$ for B mesons, are consistent with the CKM predictions

Unitarity Triangles and Experimental Tests

• The unitarity of the CKM matrix leads to unitarity triangles in the complex plane
  • The unitarity condition $V^\dagger V = 1$ implies six vanishing equations, each of which can be represented as a triangle in the complex plane
  • The most commonly studied unitarity triangle is the one related to the $B_d$ meson system, with vertices at (0,0), (1,0), and $(\bar{\rho},\bar{\eta})$
• The angles and sides of the unitarity triangles are related to CP-violating observables and can be measured experimentally
  • The angles $\alpha$, $\beta$, and $\gamma$ of the $B_d$ unitarity triangle are related to CP asymmetries in B meson decays
  • The sides of the triangles can be determined from measurements of CKM matrix elements and meson mixing parameters
• Precise measurements of the CKM matrix elements and CP-violating observables are crucial tests of the Standard Model and can provide hints of new physics beyond it
  • The global fit of all available data on CKM parameters and unitarity triangle measurements shows excellent agreement with the Standard Model predictions
  • Any significant deviations from the predicted values would indicate the presence of new sources of CP violation or flavor-changing interactions beyond the Standard Model

Experimental Evidence for the Quark Model and CKM Matrix

Discovery of Quarks and Heavy Quark Flavors

• Deep inelastic scattering experiments, such as those conducted at SLAC in the late 1960s, revealed the substructure of protons and neutrons, providing evidence for the existence of quarks
  • These experiments involved bombarding protons with high-energy electrons and measuring the angular distribution of the scattered electrons
  • The observed scaling behavior and the existence of point-like constituents within the proton were consistent with the quark model
• The discovery of the J/$\psi$ meson in 1974 confirmed the existence of the charm quark
  • The J/$\psi$ meson, with a mass of about 3.1 GeV/$c^2$, was much heavier than could be explained by the three known quarks (u, d, s) at the time
  • The charm quark was predicted by the GIM mechanism to explain the absence of flavor-changing neutral currents in kaon decays
• The observation of the Upsilon ($\Upsilon$) meson in 1977 led to the discovery of the bottom quark, completing the third generation of quarks
  • The $\Upsilon$ meson, with a mass of about 9.5 GeV/$c^2$, was even heavier than the J/$\psi$ and required a new quark flavor
  • The existence of the bottom quark was crucial for the CKM mechanism of CP violation, as it provided the necessary third generation
• The top quark, the last quark to be discovered, was observed directly at Fermilab in 1995 through top-antitop pair production in proton-antiproton collisions
  • With a mass of about 173 GeV/$c^2$, the top quark is the heaviest known elementary particle
  • The large mass of the top quark has important implications for Higgs boson physics and the hierarchy problem in the Standard Model

Measurements of CP Violation and CKM Matrix Elements

• CP violation in the neutral kaon system, first observed in 1964, provided early evidence for the CKM mechanism
  • The measured CP-violating parameters, such as $\epsilon_K$ and $\epsilon'_K$, are consistent with the CKM predictions
  • The observation of CP violation in kaon decays was a major milestone in particle physics and led to the Nobel Prize for James Cronin and Val Fitch in 1980
• The B factories, BaBar and Belle, observed CP violation in the decays of neutral B mesons in 2001
  • The measured time-dependent CP asymmetries, such as $\sin(2\beta)$ in $B^0 \to J/\psi K^0_S$ decays, agree with the CKM predictions
  • The consistency between the observed CP violation in B decays and the CKM mechanism was recognized with the Nobel Prize for Kobayashi and Maskawa in 2008
• Precision measurements of the CKM matrix elements through various weak decays have been performed at experiments like LHCb, Belle II, and BESIII
  • Semileptonic B decays, such as $B \to D^{(*)} \ell \nu$, provide measurements of $|V_{cb}|$ and $|V_{ub}|$
  • Rare kaon decays, such as $K^+ \to \pi^+ \nu \bar{\nu}$ and $K_L \to \pi^0 \nu \bar{\nu}$, are sensitive probes of $|V_{td}|$ and $|V_{ts}|$
  • Charged current interactions, such as nuclear beta decay and pion decay, determine $|V_{ud}|$ and $|V_{us}|$ with high precision
• The results from all these experiments are consistent with the unitarity of the CKM matrix and the Standard Model predictions
  • The global fit of the CKM parameters, including constraints from CP violation and rare decays, shows excellent agreement with the Standard Model
  • The success of the CKM framework in describing quark mixing and CP violation is a major triumph of the Standard Model