Neutral kaons
The system of neutral kaons is a unique laboratory for studying quantum oscillations and CP violation. In this project we concentrate on a precise measurement of the parameter \(\varepsilon\), which characterizes the violation of CP invariance.
Theory
- What are \(K^0\), \(\overline{K^0}\), \(K_S\), \(K_L\), \(K_1\), \(K_2\)?
Relate these states to one another, for example: \[\begin{aligned} |K_L\rangle &= \frac{1}{\sqrt{1+|\varepsilon|^2}}(|K_2\rangle + \varepsilon|K_1\rangle),\\ |K_S\rangle &= \frac{1}{\sqrt{1+|\varepsilon|^2}}(|K_1\rangle + \varepsilon|K_2\rangle). \end{aligned}\]
Derive the transition probabilities \(P_{K^0 \to K^0}(t)\) and \(P_{K^0 \to \bar{K}^0}(t)\) in two cases: CP symmetry conserved and CP symmetry violated.
Compute the mass difference of the neutral kaons \(\Delta m\) in the SM. Explain the essence of the GIM mechanism. Estimate the mass of the \(c\) quark.
Assuming CP symmetry is conserved, obtain the expressions for the decay rates \[\begin{aligned} R_+ & = \Gamma (K^0\to \pi^- e^+\nu_e), \\ R_- & = \Gamma (K^0\to \pi^+ e^-\overline{\nu}_e), \\ \overline{R}_+ & = \Gamma (\overline{K^0}\to \pi^+ e^-\overline{\nu}_e), \\ \overline{R}_- & = \Gamma (\overline{K^0}\to \pi^- e^+\nu_e). \end{aligned}\] Compute the decay asymmetry \[A = \frac{(R_++\overline{R}_+)-(R_-+\overline{R}_-)}{(R_++\overline{R}_+)+(R_-+\overline{R}_-)} = \frac{2e^{-(\Gamma_S+\Gamma_L)t/2}\cos\Delta m t}{e^{-\Gamma_S t}+e^{-\Gamma_L t}}.\]
An example of pseudo-data for the decay asymmetry is available at [1]. It was generated with \(\tau_S=8.954\cdot 10^{-11}\) s, \(\tau_L=5.116\cdot 10^{-8}\) s and a value of \(\Delta m\) that you have to reconstruct.
From now on we assume that CP symmetry is violated.
A beam of neutral kaons studied at \(t\gg \tau_S\) consists mainly of \(K_L\).
Show that \[\begin{aligned} \Gamma (K_L\to \pi^+ e^-\overline{\nu}_e) &\propto |\langle \overline{K^0}|K_L\rangle|^2\propto |1-\varepsilon|^2,\\ \Gamma (K_L\to \pi^- e^+\nu_e) &\propto |\langle K^0|K_L\rangle|^2\propto |1+\varepsilon|^2, \end{aligned}\] and compute the asymmetry \[A = \frac{\Gamma (K_L\to \pi^- e^+\nu_e)-\Gamma (K_L\to \pi^+ e^-\overline{\nu}_e)}{\Gamma (K_L\to \pi^- e^+\nu_e)+\Gamma (K_L\to \pi^+ e^-\overline{\nu}_e)}\] using the tabulated values.
Show how this asymmetry can be used to explain to aliens the sign of the charge of the electrons in our atoms.
Using box diagrams, show that the inequality \(\Gamma (K_L\to \pi^- e^+\nu_e)\ne\Gamma (K_L\to \pi^+ e^-\overline{\nu}_e)\) requires the CKM quark mixing matrix to be complex.
Prove that the ratio of the decay rates is \[\frac{\Gamma(K_L \to \pi^+\pi^-)}{\Gamma(K_S \to \pi^+\pi^-)} \approx |\varepsilon|^2.\]
Experimental part: pseudo-data
The table gives the data of an experiment. Reconstruct \(|\varepsilon|\), taking into account
the detection efficiency of \(\pi^+\pi^-\) in the decays of \(K_S\) and \(K_L\);
the contribution of background events.
Experimental data Decay Number of events Efficiency (%) Background (%) \(K_S \to \pi^+\pi^-\) 12 000 95 1 \(K_L \to \pi^+\pi^-\) 72 90 10
Systematic uncertainties
Assuming systematic uncertainties
of \(5\%\) in the number of background events,
of \(2\%\) in the detection efficiency,
estimate \(|\varepsilon|\) and give its statistical and systematic uncertainties.
References
1 Naumov, D. V., Pseudo-data for neutral-kaon decay asymmetry https://neutrinohit.github.io/assets/qft-exams/kaon-asymmetry-pseudodata.txt