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Part VI: Flavor Physics and the CKM Matrix

Introduction

Flavor physics studies transitions between different quark flavors, governed by the CKM (Cabibbo-Kobayashi-Maskawa) matrix. CP violation in the quark sector, discovered in 1964, is described by a complex phase in the CKM matrix.

Nobel Prize 2008: Kobayashi and Maskawa for predicting three generations of quarks

The CKM Matrix

Quark Mixing

The weak eigenstates $(d', s', b')$ are related to mass eigenstates $(d, s, b)$ by the CKM matrix:

$$ \begin{pmatrix} d' \\ s' \\ b' \end{pmatrix} = V_{\text{CKM}} \begin{pmatrix} d \\ s \\ b \end{pmatrix} = \begin{pmatrix} V_{ud} & V_{us} & V_{ub} \\ V_{cd} & V_{cs} & V_{cb} \\ V_{td} & V_{ts} & V_{tb} \end{pmatrix} \begin{pmatrix} d \\ s \\ b \end{pmatrix} $$

$V_{\text{CKM}}$ is a $3 \times 3$ unitary matrix with 4 independent parameters: 3 mixing angles and 1 CP-violating phase.

Wolfenstein Parametrization

Approximate form highlighting hierarchy ($\lambda \approx 0.22$):

$$ V_{\text{CKM}} \approx \begin{pmatrix} 1 - \frac{\lambda^2}{2} & \lambda & A\lambda^3(\rho - i\eta) \\ -\lambda & 1 - \frac{\lambda^2}{2} & A\lambda^2 \\ A\lambda^3(1-\rho-i\eta) & -A\lambda^2 & 1 \end{pmatrix} $$

CP violation encoded in complex phase $\eta \neq 0$

CP Violation

The Unitarity Triangle

Unitarity condition $V^\dagger V = I$ gives triangle relations. The most studied is:

$$ V_{ud}V^*_{ub} + V_{cd}V^*_{cb} + V_{td}V^*_{tb} = 0 $$

Rescaled triangle with vertices at $(0,0), (1,0), (\rho,\eta)$

Triangle Angles:

  • $\alpha = (89.2 \pm 3.6)°$
  • $\beta = (21.9 \pm 0.7)°$
  • $\gamma = (68.9 \pm 3.6)°$
  • Sum: $\alpha + \beta + \gamma = 180°$

Experimental Measurements

$\sin 2\beta$ from $B^0 \to J/\psi K_S$:

$\sin 2\beta = 0.699 \pm 0.017$

BaBar & Belle measurements

$|V_{ub}|$ from $B \to \pi \ell \nu$:

$|V_{ub}| = (3.70 \pm 0.10) \times 10^{-3}$

$|V_{cb}|$ from $B \to D^* \ell \nu$:

$|V_{cb}| = (39.0 \pm 0.7) \times 10^{-3}$

$B_s$ Mixing:

Constrains $|V_{ts}|$ and $|V_{td}|$

B Meson Decays

Key Processes

Tree-Level Decays:

  • $B^- \to D^0 \ell^- \bar{\nu}$: measures $|V_{cb}|$
  • $B^- \to \pi^0 \ell^- \bar{\nu}$: measures $|V_{ub}|$

Loop-Level (FCNC):

  • $B \to K^* \gamma$: radiative penguin
  • $B \to K^* \mu^+ \mu^-$: electroweak penguin
  • $B_s \to \mu^+ \mu^-$: very rare ($\text{BR} \sim 3 \times 10^{-9}$)

$B^0$-$\bar{B}^0$ Mixing:

$\Delta m_d = 0.510 \pm 0.003$ ps$^{-1}$

$\Delta m_s = 17.76 \pm 0.01$ ps$^{-1}$

Anomalies and New Physics Searches

Current Tensions

$R(D^{(*)})$ Anomaly:

Ratio of $B \to D^{(*)} \tau \nu$ to $B \to D^{(*)} \ell \nu$

$R(D) = 0.340 \pm 0.027$ (exp) vs $0.299 \pm 0.003$ (SM)

$R(D^*) = 0.295 \pm 0.014$ (exp) vs $0.258 \pm 0.005$ (SM)

Combined significance: ~3σ tension

$R(K^{(*)})$ Anomaly:

Ratio of $B \to K^{(*)} \mu^+ \mu^-$ to $B \to K^{(*)} e^+ e^-$

Hints of lepton universality violation

Recently reduced by LHCb (2022)

Key Takeaways

  • •CKM matrix: $3 \times 3$ unitary matrix describing quark flavor mixing
  • •CP violation from complex phase in CKM matrix ($\eta \neq 0$)
  • •Unitarity triangle: angles $\alpha, \beta, \gamma$ measured to ~2-4% precision
  • •B factories (BaBar, Belle) confirmed Kobayashi-Maskawa mechanism
  • •$R(D^{(*)})$ and $R(K^{(*)})$ anomalies hint at possible new physics
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