Module 3 · Weeks 5–6 · Graduate

CP Violation: From Kaons to the Jarlskog Invariant

CP violation was discovered in 1964 in neutral kaon mixing; it is encoded in the CKM and PMNS mixing matrices via one irreducible complex phase per sector. We derive the Jarlskog invariant, explain why its natural scale is too small for baryogenesis, and review the experimental search for beyond-SM CP violation in B mesons, neutrinos, and electric dipole moments.

1. The Discrete Symmetries C, P, T

Three discrete symmetries act on quantum fields:

Parity P: \(\vec x \to -\vec x\), spatial reflection.
Charge conjugation C: particle \(\leftrightarrow\) antiparticle; all internal quantum numbers flip sign.
Time reversal T: \(t\to -t\), reverses momenta and spins; antiunitary.

QED and QCD are invariant under each individually. The weak interaction violates P maximally (Wu 1957) and C maximally (charged W couples only to left-handed fermions). But the combination CP was initially thought exact.

In 1964, Cronin and Fitch observed \(K_L \to \pi^+\pi^-\) at a rate \(2\times 10^{-3}\) relative to the dominant \(K_L \to 3\pi\). This small but non-zero decay proved CP is violated, earning the 1980 Nobel Prize.

2. The Neutral Kaon System

Strong-interaction eigenstates \(K^0 = |d\bar s\rangle\) and \(\bar K^0 = |\bar d s\rangle\) mix via the weak interaction box diagram. CP eigenstates (if CP were exact):

\[|K_1\rangle = \frac{1}{\sqrt 2}\bigl(|K^0\rangle + |\bar K^0\rangle\bigr), \quad CP|K_1\rangle = +|K_1\rangle\]\[|K_2\rangle = \frac{1}{\sqrt 2}\bigl(|K^0\rangle - |\bar K^0\rangle\bigr), \quad CP|K_2\rangle = -|K_2\rangle\]

Observed mass eigenstates \(K_S\), \(K_L\) are not exactly \(K_{1,2}\):

The measured CP-violation parameter is \(|\epsilon_K| = (2.228 \pm 0.011)\times 10^{-3}\)and there is also direct CP violation \(\epsilon'/\epsilon \approx 1.7\times10^{-3}\)(NA48 and KTeV).

3. The CKM Matrix

In the Standard Model with three generations, the charged-current weak interaction connects up-type and down-type quarks via the \(3\times 3\) unitary Cabibbo-Kobayashi-Maskawa matrix

\[V_{CKM} = \begin{pmatrix} V_{ud} & V_{us} & V_{ub} \\ V_{cd} & V_{cs} & V_{cb} \\ V_{td} & V_{ts} & V_{tb} \end{pmatrix}.\]

A general \(3\times3\) unitary matrix has 9 real parameters. Five of them can be absorbed into redefinitions of the quark phases (9 quark fields minus 1 overall phase = 5). That leaves 3 angles and 1 irreducible CP-violating phase — the source of all SM CP violation in the quark sector (Kobayashi & Maskawa 1973, Nobel 2008).

Counting parameters

\(U(3)\) has 9 generators, so 9 real parameters for a \(3\times 3\) unitary matrix
Orthogonal part \(O(3)\): 3 Euler angles
Remaining: 6 phases; 5 can be absorbed into quark-field phase redefinitions
\(\Rightarrow\) 3 angles + 1 phase (CP-violating) for three generations
For N generations: \(N(N-1)/2\) angles and \((N-1)(N-2)/2\) CP phases. Need \(N \geq 3\) for any CP violation.

Wolfenstein parametrization (using the small parameter \(\lambda = \sin\theta_{12} \approx 0.225\)):

\[V_{CKM} \approx \begin{pmatrix} 1-\lambda^2/2 & \lambda & A\lambda^3(\rho - i\eta) \\ -\lambda & 1-\lambda^2/2 & A\lambda^2 \\ A\lambda^3(1-\rho-i\eta) & -A\lambda^2 & 1 \end{pmatrix}.\]

Measured values: \(A = 0.811\), \(\bar\rho = 0.155\), \(\bar\eta = 0.357\) (CKMfitter 2022).

4. The Jarlskog Invariant

All CP violation in the SM quark sector is encoded in a single real, rephasing-invariant quantity. For any four indices \((i,j,k,l)\) with \(i\neq k\), \(j \neq l\):

\[\mathrm{Im}\!\left[V_{ij} V_{kl} V_{il}^* V_{kj}^*\right] = J \sum_{m,n} \varepsilon_{ikm}\,\varepsilon_{jln}.\]

\(J\) is the same for every choice of indices (up to sign). Using \((i,k) = (u,c),\; (j,l) = (s,b)\):

\[J_{CP} = \mathrm{Im}\!\left[V_{us} V_{cb} V_{ub}^* V_{cs}^*\right].\]

In the Wolfenstein parametrization: \(J \approx A^2 \lambda^6 \eta \approx 3.1 \times 10^{-5}\). This is the maximum CP-violating effect in the SM quark sector.

Geometric interpretation: the unitarity triangle

Unitarity of CKM gives \(\sum_q V_{qi} V_{qj}^* = \delta_{ij}\). For \((i,j) = (d,b)\):

\[V_{ud} V_{ub}^* + V_{cd} V_{cb}^* + V_{td} V_{tb}^* = 0.\]

This expresses a triangle in the complex plane. Its area is \(J/2\). The three angles \(\alpha, \beta, \gamma\) are measured at B factories to verify SM unitarity (or find new physics).

5. Diagram: CKM Unitarity Triangle

The unitarity triangle is closed by three CKM matrix elements. Its non-zero area signals CP violation; the three angles \(\alpha,\beta,\gamma\) can all be measured independently at B factories, providing the strongest test of the Kobayashi-Maskawa mechanism.

6. Lepton-Sector CP Violation: PMNS Matrix

Neutrino oscillations require neutrino masses; flavor eigenstates \((\nu_e,\nu_\mu,\nu_\tau)\) are related to mass eigenstates \((\nu_1,\nu_2,\nu_3)\) via the Pontecorvo-Maki-Nakagawa-Sakata matrix:

\[U_{PMNS} = \begin{pmatrix} c_{12}c_{13} & s_{12}c_{13} & s_{13}e^{-i\delta_{CP}} \\ -s_{12}c_{23} - c_{12}s_{13}s_{23}e^{i\delta} & c_{12}c_{23} - s_{12}s_{13}s_{23}e^{i\delta} & c_{13}s_{23} \\ s_{12}s_{23} - c_{12}s_{13}c_{23}e^{i\delta} & -c_{12}s_{23} - s_{12}s_{13}c_{23}e^{i\delta} & c_{13}c_{23} \end{pmatrix}.\]

If neutrinos are Majorana fermions, two additional CP-violating phases \(\alpha_{21},\alpha_{31}\)appear in a diagonal matrix. For baryogenesis via leptogenesis, these Majorana phases matter.

T2K and NOvA measure \(\delta_{CP}\). The combined 2023 fit prefers \(\delta_{CP} \approx -\pi/2\) (maximal CP violation in the lepton sector) but with large uncertainties. DUNE and Hyper-K will measure it at the 10\u00B0 level.

7. Why SM CP Violation is Too Small for Baryogenesis

The Jarlskog invariant \(J_{CP} \sim 3\times 10^{-5}\) looks large enough, but physical CP-violating observables in the SM always come multiplied by a dimensionalsuppression from quark masses. At the electroweak temperature \(T \sim 100\) GeV:

\[\delta_{CP} \sim \frac{J_{CP} \prod_{i<j}(m_{u_i}^2-m_{u_j}^2)(m_{d_i}^2-m_{d_j}^2)}{T^{12}} \sim 10^{-20}.\]

This is 10 orders of magnitude smaller than \(\eta_{obs} \sim 10^{-10}\). Therefore BSM CP violation is required — in extended Higgs sectors, supersymmetry, right-handed neutrinos, or elsewhere.

Simulation: CKM Matrix, Jarlskog, and Unitarity Triangle

The following simulation constructs the CKM matrix from measured parameters, numerically verifies unitarity, computes the Jarlskog invariant in multiple ways, and draws the unitarity triangle along with a comparison of Pythagorean-style uncertainty ellipses from global fits.

Python

script.py173 lines

import numpy as np
import matplotlib
matplotlib.use('Agg')
import matplotlib.pyplot as plt
import matplotlib.gridspec as gridspec

plt.rcParams['font.family'] = 'monospace'
BG='#0a0a1a'; INDIGO='#818cf8'; PURPLE='#c4b5fd'; WARN='#ef4444'; GOLD='#f59e0b'; GREEN='#34d399'

# ---------- Wolfenstein parameters (PDG 2022) ----------
lam    = 0.22500
A      = 0.826
rho_b  = 0.159       # rho-bar
eta_b  = 0.348       # eta-bar

# Convert to rho, eta
rho = rho_b / (1 - lam**2/2)
eta = eta_b / (1 - lam**2/2)

# Build CKM matrix (Wolfenstein expansion to O(lambda^4))
V = np.zeros((3,3), dtype=complex)
V[0,0] = 1 - lam**2/2 - lam**4/8
V[0,1] = lam
V[0,2] = A * lam**3 * (rho - 1j*eta)
V[1,0] = -lam + A**2*lam**5*(1 - 2*(rho + 1j*eta))/2
V[1,1] = 1 - lam**2/2 - lam**4*(1 + 4*A**2)/8
V[1,2] = A*lam**2
V[2,0] = A * lam**3 * (1 - rho - 1j*eta)
V[2,1] = -A*lam**2 + A*lam**4*(1 - 2*(rho + 1j*eta))/2
V[2,2] = 1 - A**2 * lam**4 / 2

print('=== CKM Matrix (Wolfenstein to O(lambda^4)) ===')
labels = [['Vud','Vus','Vub'],['Vcd','Vcs','Vcb'],['Vtd','Vts','Vtb']]
for i in range(3):
    for j in range(3):
        print(f'  {labels[i][j]:>4} = {V[i,j].real:+.5f} {V[i,j].imag:+.5f}i  |V|={abs(V[i,j]):.5f}')
    print()

# ---------- Verify unitarity ----------
UUt = V @ V.conj().T
print('=== V V-dagger (should be identity) ===')
for i in range(3):
    row = ' '.join(f'{abs(UUt[i,j]):.4f}' for j in range(3))
    print(f'  {row}')

# ---------- Jarlskog invariant (multiple forms) ----------
J1 = np.imag(V[0,1] * V[1,2] * np.conj(V[0,2]) * np.conj(V[1,1]))
J2 = np.imag(V[0,0] * V[1,1] * np.conj(V[0,1]) * np.conj(V[1,0]))
J3 = np.imag(V[1,0] * V[2,2] * np.conj(V[1,2]) * np.conj(V[2,0]))
J_wolf = A**2 * lam**6 * eta
print('\n=== Jarlskog invariant (rephasing-invariant) ===')
print(f'  J_1 = Im[Vus Vcb Vub* Vcs*]  = {J1:+.4e}')
print(f'  J_2 = Im[Vud Vcs Vus* Vcd*]  = {J2:+.4e}')
print(f'  J_3 = Im[Vcd Vtb Vcb* Vtd*]  = {J3:+.4e}')
print(f'  J_Wolfenstein = A^2 lam^6 eta = {J_wolf:+.4e}')

# ---------- Unitarity triangle ----------
# Normalize by Vcd Vcb*
V_ud_Vub_star = V[0,0] * np.conj(V[0,2])
V_cd_Vcb_star = V[1,0] * np.conj(V[1,2])
V_td_Vtb_star = V[2,0] * np.conj(V[2,2])
norm = V_cd_Vcb_star

A_vert = V_ud_Vub_star / norm
B_vert = 0 + 0j
C_vert = -V_td_Vtb_star / norm  # since (unit triangle) A + (something) = 1

# Actually standard: (V_ud V_ub* + V_cd V_cb* + V_td V_tb*)/|V_cd V_cb*|
sum_check = V_ud_Vub_star + V_cd_Vcb_star + V_td_Vtb_star
print(f'\nUnitarity sum V_ud V_ub* + V_cd V_cb* + V_td V_tb* = {sum_check:.3e} (should be ~0)')

# Triangle in (rho-bar, eta-bar) plane
A_re, A_im = A_vert.real, A_vert.imag

print(f'\n=== Unitarity triangle vertices ===')
print(f'  B = (0, 0)')
print(f'  C = (1, 0)')
print(f'  A = ({A_re:.3f}, {A_im:.3f})')

# Compute angles
def angle_deg(v1, v2):
    dot = np.real(np.conj(v1)*v2)
    cross = np.imag(np.conj(v1)*v2)
    return np.degrees(np.arctan2(cross, dot))

alpha = angle_deg(-V_ud_Vub_star/norm, V_td_Vtb_star/norm)
beta  = angle_deg(-V_cd_Vcb_star/norm, -V_td_Vtb_star/norm)
gamma = angle_deg(-V_ud_Vub_star/norm, -V_cd_Vcb_star/norm)
# normalise to [0, 180]
alpha, beta, gamma = abs(alpha) % 180, abs(beta) % 180, abs(gamma) % 180
print(f'  alpha = {alpha:.1f}\u00B0')
print(f'  beta  = {beta:.1f}\u00B0')
print(f'  gamma = {gamma:.1f}\u00B0')
print(f'  sum   = {alpha+beta+gamma:.1f}\u00B0  (SM expects 180\u00B0)')

# ---------- Plot ----------
fig = plt.figure(figsize=(14, 10), facecolor=BG)
fig.suptitle('CKM Matrix, Jarlskog Invariant, and the Unitarity Triangle',
             color='white', fontsize=16, fontweight='bold', y=0.98)
gs = gridspec.GridSpec(2, 2, figure=fig, hspace=0.40, wspace=0.32)

# Heatmap of |V_ij|
ax1 = fig.add_subplot(gs[0, 0], facecolor=BG)
abs_V = np.abs(V)
im = ax1.imshow(abs_V, cmap='plasma', vmin=0, vmax=1)
for i in range(3):
    for j in range(3):
        ax1.text(j, i, f'{abs_V[i,j]:.4f}', ha='center', va='center',
                 color='white' if abs_V[i,j]<0.7 else 'black', fontsize=10)
ax1.set_xticks([0,1,2], ['d','s','b'], color=PURPLE)
ax1.set_yticks([0,1,2], ['u','c','t'], color=PURPLE)
ax1.set_title('|V_CKM|: magnitudes of matrix elements', color='white', fontsize=11)

# Unitarity triangle
ax2 = fig.add_subplot(gs[0, 1], facecolor=BG)
ax2.plot([0, A_re], [0, A_im], color=INDIGO, lw=2)
ax2.plot([0, 1], [0, 0], color=INDIGO, lw=2)
ax2.plot([1, A_re], [0, A_im], color=INDIGO, lw=2)
ax2.fill([0, A_re, 1], [0, A_im, 0], color=PURPLE, alpha=0.15)
ax2.scatter([0, 1, A_re], [0, 0, A_im], color=[GREEN, GREEN, GOLD], s=80, zorder=5)
ax2.annotate('B(0,0)', (0, 0), xytext=(-40, -15), textcoords='offset points', color=GREEN, fontsize=10)
ax2.annotate('C(1,0)', (1, 0), xytext=( 10, -15), textcoords='offset points', color=GREEN, fontsize=10)
ax2.annotate(f'A=({A_re:.3f},{A_im:.3f})', (A_re, A_im), xytext=(10, 10), textcoords='offset points', color=GOLD, fontsize=10)
ax2.text(0.5, -0.05, '\u03B3', color='white', fontsize=13, ha='center')
ax2.text(A_re-0.05, A_im/2, '\u03B1', color='white', fontsize=13, ha='right')
ax2.text(0.7, 0.1, '\u03B2', color='white', fontsize=13)
ax2.set_xlim(-0.1, 1.1); ax2.set_ylim(-0.1, 0.5)
ax2.axhline(0, color='white', lw=0.3)
ax2.axvline(0, color='white', lw=0.3)
ax2.set_xlabel('\u03C1\u0305', color=PURPLE); ax2.set_ylabel('\u03B7\u0305', color=PURPLE)
ax2.set_title('Unitarity triangle', color='white', fontsize=11)
ax2.tick_params(colors=PURPLE)
ax2.grid(True, alpha=0.15, color=PURPLE)
for sp in ax2.spines.values(): sp.set_color('#312e81')

# Angles comparison: SM prediction vs measurement
ax3 = fig.add_subplot(gs[1, 0], facecolor=BG)
meas  = [84.9, 22.14, 72.1]
errs  = [2.0,  0.69,  5.8]
comp  = [alpha, beta, gamma]
xlabs = [r'\u03B1', r'\u03B2', r'\u03B3']
x = np.arange(3)
ax3.bar(x-0.2, comp, 0.4, label='CKM (Wolfenstein)', color=INDIGO, alpha=0.85)
ax3.bar(x+0.2, meas, 0.4, yerr=errs, label='CKMfitter 2022', color=PURPLE, alpha=0.85, capsize=5)
ax3.set_xticks(x); ax3.set_xticklabels(xlabs, color=PURPLE, fontsize=14)
ax3.set_ylabel('angle [\u00B0]', color=PURPLE)
ax3.set_title('Unitarity triangle angles: SM vs measurement', color='white', fontsize=11)
ax3.legend(fontsize=9, facecolor='#0f172a', labelcolor='white', edgecolor='#312e81')
ax3.tick_params(colors=PURPLE)
ax3.grid(True, alpha=0.15, color=PURPLE)
for sp in ax3.spines.values(): sp.set_color('#312e81')

# CP-violating suppression for SM baryogenesis
ax4 = fig.add_subplot(gs[1, 1], facecolor=BG)
Tvals = np.logspace(1, 3, 200)
mt,mc,mu = 173, 1.27, 0.0022
mb,ms,md = 4.18, 0.093, 0.0047
num = (mt**2-mc**2)*(mc**2-mu**2)*(mt**2-mu**2)*(mb**2-ms**2)*(ms**2-md**2)*(mb**2-md**2)
dCP = J1 * num / Tvals**12
ax4.loglog(Tvals, np.abs(dCP), color=INDIGO, lw=2.5)
ax4.axhline(1e-10, color=GOLD, lw=1.5, ls='--', label='Required for baryogenesis')
ax4.fill_between(Tvals, 1e-30, np.abs(dCP), color=INDIGO, alpha=0.15)
ax4.set_xlabel('temperature T [GeV]', color=PURPLE)
ax4.set_ylabel(r'SM CP signal \u03B4_CP', color=PURPLE)
ax4.set_title('SM CP effect vs required (10 orders short)', color='white', fontsize=11)
ax4.legend(fontsize=9, facecolor='#0f172a', labelcolor='white', edgecolor='#312e81')
ax4.tick_params(colors=PURPLE)
ax4.grid(True, alpha=0.15, color=PURPLE)
for sp in ax4.spines.values(): sp.set_color('#312e81')

plt.savefig('output.png', dpi=110, bbox_inches='tight', facecolor=BG)
print('\nSaved output.png')

Click Run to execute the Python code

Code will be executed with Python 3 on the server

8. B-Meson CP Violation: BaBar, Belle, LHCb

B mesons are the cleanest laboratory for testing the CKM mechanism at the 1% level. The dominant observables are:

Indirect CP violation in B \(^0\) mixing: \(\sin 2\beta\) from \(B^0 \to J/\psi K_S\), measured to be 0.699 \u00B1 0.017.
Direct CP violation: rate asymmetries in \(B \to K\pi\) and \(B\to \pi\pi\), at the few-percent level.
CP violation in mixing: \(|q/p| - 1 < 10^{-3}\) constrains semileptonic asymmetries.
B \(_s\) mixing phase: \(\phi_s\) small in SM, sensitive to new physics.

Running at LHCb and Belle II, and at future colliders, continues to stress-test the CKM mechanism and search for BSM CP phases.

9b. Neutrino Mixing Parameters (NuFit 2023)

Current best-fit values from the global NuFit analysis of solar, atmospheric, reactor, and accelerator neutrino experiments:

Mixing angles

\(\sin^2\theta_{12}\) = 0.307 \u00B1 0.013 (solar)
\(\sin^2\theta_{23}\) = 0.545 \u00B1 0.021 (atm)
\(\sin^2\theta_{13}\) = 0.02203 \u00B1 0.00056 (reactor)

Mass splittings

\(\Delta m^2_{21}\) = 7.42 \u00D7 10 \(^{-5}\) eV \(^2\)
\(|\Delta m^2_{31}|\) = 2.51 \u00D7 10 \(^{-3}\) eV \(^2\)
Hierarchy: preferred NH (normal)

The CP phase \(\delta_{CP}\) is currently at \(-1.52^{+0.26}_{-0.32}\) rad (NH, NuFit), with T2K+NOvA tension. Upcoming Hyper-K and DUNE will pin it down to ∼10\u00B0 precision.

9c. Axions and the PQ Mechanism

The Peccei-Quinn solution to the strong CP problem introduces a global U(1)_PQ symmetry spontaneously broken at a scale \(f_a\), giving rise to a pseudo-Goldstone boson — the axion — with a potential generated by QCD instantons. The axion dynamically sets \(\bar\theta = 0\).

Axion relic density \(\Omega_a \sim 0.1\,(f_a/10^{12}~\mathrm{GeV})^{7/6}\theta_i^2\) makes axions a viable dark matter candidate. Experiments like ADMX, HAYSTAC, and MADMAX search for axion-photon conversion in strong magnetic fields.

In some scenarios, the axion's rolling during inflation can also contribute to baryogenesis (“axionic baryogenesis”). This links CP violation, the strong CP problem, dark matter, and baryogenesis in a single framework.

9. The Strong CP Problem

QCD admits a CP-violating term in its Lagrangian:

\[\mathcal L_{\theta} = \theta\,\frac{g_s^2}{32\pi^2}\,G_{\mu\nu}^a\tilde G^{a\,\mu\nu}.\]

Combined with the quark-mass phase \(\arg\det M_q\), the physical parameter is \(\bar\theta = \theta + \arg\det M_q\). A nonzero \(\bar\theta\)generates a neutron EDM \(d_n \sim e\bar\theta\,(m_q/\Lambda_{QCD})\). The current limit \(|d_n| < 1.8\times10^{-26}\) e\u00B7cm implies

\[|\bar\theta| \lesssim 10^{-10}.\]

Why is \(\bar\theta\) so small? The leading solution, the Peccei-Quinn mechanism (1977), introduces a new U(1)_PQ symmetry that dynamically relaxes \(\bar\theta\) to zero via an axion field. The strong CP problem is related to baryogenesis because axion dynamics can modify the story in the very early universe.

References

Christenson, J. H., Cronin, J. W., Fitch, V. L. & Turlay, R., “Evidence for the 2\u03C0 decay of the K_2\u2070 meson”, Phys. Rev. Lett. 13, 138 (1964).
Kobayashi, M. & Maskawa, T., “CP-violation in the renormalizable theory of weak interaction”, Prog. Theor. Phys. 49, 652 (1973).
Jarlskog, C., “Commutator of the quark mass matrices in the standard electroweak model and a measure of maximal CP nonconservation”, Phys. Rev. Lett. 55, 1039 (1985).
Wolfenstein, L., “Parametrization of the Kobayashi-Maskawa matrix”, Phys. Rev. Lett. 51, 1945 (1983).
Particle Data Group, “Review of Particle Physics”, Prog. Theor. Exp. Phys. 2022, 083C01 (2022).
T2K Collaboration, “Constraint on the matter-antimatter symmetry-violating phase in neutrino oscillations”, Nature 580, 339 (2020).
Huet, P. & Sather, E., “Electroweak baryogenesis and standard model CP violation”, Phys. Rev. D 51, 379 (1995).

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