Quantum Field Theory Course

Total: 440+ pages covering classical field theory through the Standard Model

Part VIII: Standard Model • Chapter 3

Quark Sector & CKM Matrix

Flavor mixing, CP violation, and the origin of matter

The Flavor Puzzle

Why are there three generations of quarks?

1st Generation

u, d
~MeV masses

2nd Generation

c, s
~GeV masses

3rd Generation

t, b
~100 GeV masses

Each generation has identical gauge interactions, differing only in Yukawa couplings (masses). The weak force, however, mixes these generations through the CKM matrix.

1. Quark Masses

Quarks get mass from Yukawa couplings to the Higgs:

ℒYukawa = -Yiju Q̄Li Φ̃ uRj - Yijd Q̄Li Φ dRj + h.c.

where Yu and Yd are 3×3 complex matrices (27 complex parameters each!).

QuarkMass (MS̄, 2 GeV)Yukawa y = √2 m/v
up (u)2.2 MeV1.3 × 10-5
down (d)4.7 MeV2.7 × 10-5
strange (s)95 MeV5.5 × 10-4
charm (c)1.28 GeV7.4 × 10-3
bottom (b)4.18 GeV2.4 × 10-2
top (t)173 GeV1.0 (≈ 1!)

Remarkable fact:

The top Yukawa yt ≈ 1 is O(1)! The top quark has a "natural" coupling to the Higgs. All other fermions are mysteriously suppressed by factors of 10⁻² to 10⁻⁵. Why?

2. The CKM Matrix

Origin: Mass Basis vs Weak Basis

The Yukawa matrices Yu and Yd are not diagonal in the weak interaction basis. We diagonalize them via bi-unitary transformations:

VLu† Yu VRu = diag(yu, yc, yt)
VLd† Yd VRd = diag(yd, ys, yb)

In the mass eigenstate basis, the charged current W interaction becomes:

ℒCC = (g/√2) ūLi γμ Vij dLj Wμ+ + h.c.

where the CKM matrix is:

VCKM = VLu† VLd

This 3×3 unitary matrix describes how mass eigenstates (u, c, t) couple to (d, s, b) via W± exchange.

CKM Matrix Structure

The standard parametrization:

V = ⎡ Vud Vus Vub ⎤
    ⎢ Vcd Vcs Vcb ⎥
    ⎣ Vtd Vts Vtb ⎦

A general 3×3 unitary matrix has 9 parameters:

  • • 9 real parameters - 3 normalization conditions - 5 unphysical phases = 4 physical parameters
  • • 3 mixing angles: θ12, θ23, θ13
  • • 1 complex phase: δCP (source of CP violation!)

Wolfenstein Parametrization:

Expansion in powers of λ = sin θ12 ≈ 0.22 (Cabibbo angle):

V ≈ ⎡ 1-λ²/2 λ Aλ³(ρ-iη) ⎤
    ⎢ -λ 1-λ²/2 Aλ² ⎥ + O(λ⁴)
    ⎣ Aλ³(1-ρ-iη) -Aλ² 1 ⎦

Parameters: λ ≈ 0.225, A ≈ 0.81, ρ̄ ≈ 0.16, η̄ ≈ 0.35

Experimental Values

Element|Value|Interpretation
|Vud|0.974u ↔ d: nuclear β decay
|Vus|0.225u ↔ s: Cabibbo-suppressed
|Vub|0.004u ↔ b: highly suppressed
|Vcd|0.221c ↔ d: Cabibbo-suppressed
|Vcs|0.975c ↔ s: Cabibbo-favored
|Vcb|0.041c ↔ b: B meson decays
|Vtd|0.009t ↔ d: highly suppressed
|Vts|0.040t ↔ s: Bs mixing
|Vtb|0.999t ↔ b: top decay (almost 100%!)

Key Pattern:

Diagonal elements ≈ 1, off-diagonal suppressed. Transitions within one generation are favored, between adjacent generations are suppressed by ~0.2, and between 1st-3rd generation are tiny (~10⁻³).

3. CP Violation

The Jarlskog Invariant

CP violation in the SM arises from the complex phase δCP in VCKM. The unique measure is:

J = Im[Vus Vcb Vub* Vcs*] ≈ 3 × 10-5

This single number quantifies all CP violation in quark sector. It appears in processes like:

  • • K0 - K̄0 mixing: εK parameter in kaon decays
  • • B0 - B̄0 mixing: observed asymmetries in B factories (BaBar, Belle)
  • • D0 - D̄0 mixing: charm sector (very small)

Unitarity Triangles

Unitarity of VCKM gives 6 orthogonality relations. The most studied is:

Vud Vub* + Vcd Vcb* + Vtd Vtb* = 0

This forms a triangle in the complex plane (rescaled by VcdVcb*):

The Unitarity Triangle:

Apex (ρ̄, η̄):
Determined by B decays
Angle α:
B → ππ, ρρ
Angle β:
B → J/ψ KS

The triangle has been measured from all angles—remarkable consistency confirms the CKM paradigm!

Matter-Antimatter Asymmetry

Sakharov's conditions for baryogenesis require CP violation. The CKM phase provides this, but:

Problem:

CKM CP violation is too small to explain the observed baryon asymmetry ηB = (nB - nB̄)/nγ ≈ 6 × 10-10. We need new sources of CP violation beyond the SM (e.g., leptogenesis, SUSY).

4. Flavor-Changing Processes

FCNC Suppression

Flavor-Changing Neutral Currents (FCNC) like s → d + γ are absent at tree level in the SM. The GIM mechanism (Glashow-Iliopoulos-Maiani) explains this:

  • • Z and γ couple to mass eigenstates diagonally (no VCKM for neutral currents)
  • • FCNC only appear at loop level via W boson exchange
  • • Suppressed by (mc² - mu²)/MW² ~ 10-6

Key Processes:

B0 - B̄0 oscillations:

Box diagrams with top quark loops. Mixing frequency ΔmB ∝ |VtbVtd|². Observed with ~ps-1 frequency at B factories.

KL → π+π- vs KL → π0π0:

Direct CP violation parameter ε'/ε ≈ 10-3. Measured by NA48, KTeV experiments.

b → sγ penguin diagrams:

B → K*γ decay. Sensitive to new physics in top/W loops. Branching ratio ~10-5.

Summary

  • ✓ 6 quarks in 3 generations: (u,d), (c,s), (t,b) with vastly different masses
  • ✓ CKM matrix VCKM: 3×3 unitary, 4 parameters (3 angles + 1 phase)
  • ✓ Flavor mixing: W± couples mass eigenstates via Vij
  • ✓ CP violation: Complex phase δCP, Jarlskog invariant J ~ 10-5
  • ✓ Unitarity triangle: Overdetermined by B decay measurements (ρ̄, η̄)
  • ✓ FCNC suppression: GIM mechanism, only at loop level
  • ✓ Top quark special: Vtb ≈ 1, decays before hadronizing (Γt ~ 1.5 GeV)

Further Resources

  • • PDG Review - "The CKM Quark-Mixing Matrix"
  • • Bigi & Sanda - CP Violation (Cambridge, 2000)
  • • Branco, Lavoura, Silva - CP Violation (Oxford, 1999)
  • • CKMfitter Group - http://ckmfitter.in2p3.fr (live unitarity triangle fits)