The neutral kaon system: from discovery to CP violation

Rochester and Butler’s 1947 cloud-chamber photographs of forked “V0” tracks were the first examples of the “strange particles,” among them the neutral kaon – a genuinely new two-state system whose mixing would, seventeen years later, supply the first evidence for CP violation. By the early 1960s, CP symmetry predicted that the long-lived neutral kaon \(K_L\), being CP-odd, could not decay to two pions; Cronin and Fitch (1964) found that it does, at a rate of about two parts in a thousand. This example models the neutral-kaon system as the Wigner-Weisskopf two-state mixing problem meson_decay_rates_cp_eigenstate() implements, and reproduces the decay-rate asymmetry cp_asymmetry() that a real measurement of \(\Delta m\) and CP violation is built from.

import matplotlib.pyplot as plt
import numpy as np

from physicskit.particle.electroweak import cp_asymmetry, meson_decay_rates_cp_eigenstate
from physicskit.particle.visualizers import animate_cp_asymmetry

The two very different lifetimes Rochester and Butler’s kaon system has

K_short and K_long decay at very different rates – the “unexpectedly long lifetime” that gave the strange particles their name in the first place, in units of the K_S decay width here.

delta_m = 0.477  # in units of Gamma_S, close to the real K0 system's value
gamma_s = 1.0
gamma_l = gamma_s / 579.0  # K_L lives roughly 579x longer than K_S

t = np.linspace(0.0, 20.0, 1000)  # in units of 1/Gamma_S (K_S lifetimes)

No CP violation: identical K0 and K0-bar decay rates

gamma_meson_0, gamma_mesonbar_0 = meson_decay_rates_cp_eigenstate(t, delta_m, gamma_s, gamma_l, epsilon=0.0)
print(f"epsilon=0 (CP conserved): max |Gamma(K0) - Gamma(K0bar)| = {np.max(np.abs(gamma_meson_0 - gamma_mesonbar_0)):.2e} (identical, as CP demands)")

epsilon=0 (CP conserved): max |Gamma(K0) - Gamma(K0bar)| = 0.00e+00 (identical, as CP demands)

Cronin and Fitch’s actual result: a small but real CP-violating admixture

The real K0 system’s CP-violation parameter is tiny, \(|\epsilon| \sim 2.2\times10^{-3}\).

epsilon = 2.2e-3 * np.exp(1j * np.radians(43.5))  # the real K0 system's measured |epsilon| and phase
gamma_meson, gamma_mesonbar = meson_decay_rates_cp_eigenstate(t, delta_m, gamma_s, gamma_l, epsilon)
A = cp_asymmetry(t, delta_m, gamma_s, gamma_l, epsilon)

print(f"epsilon={abs(epsilon):.4f} (Cronin-Fitch's actual result): decay rates now genuinely differ")
print(f"asymmetry A(t): oscillates with amplitude up to {np.max(np.abs(A)):.4f}, inside a decaying envelope")

fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(11, 4.2))
ax1.semilogy(t, gamma_meson, color="steelblue", label=r"$\Gamma(K^0\to f)$")
ax1.semilogy(t, gamma_mesonbar, color="firebrick", label=r"$\Gamma(\bar K^0\to f)$")
ax1.set_xlabel(r"proper time $t$ (units of $1/\Gamma_S$)")
ax1.set_ylabel("decay rate to a CP eigenstate")
ax1.set_title("K0 vs K0-bar decay rates: nearly, but not exactly, equal")
ax1.legend(fontsize=8)

ax2.plot(t, A, color="darkorange")
ax2.axhline(0, color="0.6", lw=0.8)
ax2.set_xlabel(r"proper time $t$ (units of $1/\Gamma_S$)")
ax2.set_ylabel("CP asymmetry A(t)")
ax2.set_title(r"A nonzero asymmetry: CP violation, $\varepsilon\neq0$")
fig.tight_layout()

epsilon=0.0022 (Cronin-Fitch's actual result): decay rates now genuinely differ
asymmetry A(t): oscillates with amplitude up to 0.6897, inside a decaying envelope

Building up the measured asymmetry over time

anim = animate_cp_asymmetry(t, delta_m, gamma_s, gamma_l, epsilon)

plt.show()