[Paper Review] New Results on Rare and Forbidden Semileptonic $K^+$ Decays

P. Truoel|ArXiv.org|Dec 5, 2000

Particle physics theoretical and experimental studies3 citations

TL;DR

This paper presents high-precision measurements of rare and forbidden semileptonic $K^+$ decays using the E865 experiment at Brookhaven AGS, achieving improved sensitivity to lepton flavour violation in $K^+ \to \pi^+\mu^+e^-$ and extracting a new, precise value for the $s$-wave $\pi\pi$ scattering length from $K_{e4}$ decays. The results provide strong constraints on chiral perturbation theory and confirm next-to-leading-order predictions with high accuracy.

ABSTRACT

Experiment E865 at the Brookhaven AGS was set up primarily to search for the lepton flavour violating decay $K^+ o π^+μ^+e^-$ ($K_{πμe}$) with high sensitivity. The flexibility of the apparatus allowed also to obtain more than an order of magnitude larger than previously available event samples on the following decay modes: $ π^+e^+e^- (K_{πee})$, $π^+μ^+μ^- (K_{πμμ})$, $π^+π^-e^+ν_e (K_{e4})$, $μ^+e^+e^-ν_μ$, and $e^+e^+e^-ν_e$. The report focusses on the $K_{πμe}$ results and those on other lepton flavour violating decays as well as the $K_{e4}$ data, from which a new, quite precise value for the s-wave $ππ$ scattering length can be deduced.

Motivation & Objective

• To search for lepton flavour violating decays such as $K^+ \to \pi^+\mu^+e^-$ with improved sensitivity beyond previous limits.
• To measure rare semileptonic decays like $K_{\pi ee}$, $K_{\pi\mu\mu}$, and $K_{e4}$ with high-statistics samples.
• To extract the $s$-wave $\pi\pi$ scattering length $a_0^0$ from $K_{e4}$ decays using a model-independent analysis.
• To test predictions of chiral perturbation theory (ChPT) using form factor and phase shift measurements from $K_{e4}$ decays.
• To reduce systematic biases in form factor extraction by applying event-by-event matrix element corrections in the fit.

Proposed method

• The E865 spectrometer at Brookhaven AGS detected $K^+$ decays in a 5 m decay volume using a 6 GeV/$c$ beam with high particle identification and momentum resolution.
• Charged particles were momentum-analyzed using two dipole magnets and tracked with four-plane proportional chambers, achieving $\sigma_p/p^2 \approx 0.003$.
• Electron identification was performed using four gas-filled Cherenkov counters and a Shashlyk-type electromagnetic calorimeter.
• Muon identification was enhanced using a muon stack with 12 planes of proportional tubes behind iron plates.
• The trigger required three charged particles with $e^\pm$ identification and topological constraints, enabling high-statistics data collection on multiple decay modes.
• A $\chi^2$ minimization procedure compared observed event distributions in 28,800 kinematic bins to Monte Carlo simulations, with form factors and phase shifts extracted via event-by-event matrix element reweighting.

Experimental results

Research questions

• RQ1 What is the branching ratio of the lepton flavour violating decay $K^+ \to \pi^+\mu^+e^-$, and how does it compare to the Standard Model prediction?
• RQ2 Can the $K_{e4}$ decay mode $K^+ \to \pi^+\pi^- e^+ \nu_e$ be used to extract the $s$-wave $\pi\pi$ scattering length with high precision?
• RQ3 How well do the measured form factors and phase shifts in $K_{e4}$ decay agree with next-to-leading-order chiral perturbation theory predictions?
• RQ4 What is the impact of the universal band constraint on the $a_0^0$-$a_0^2$ scattering length plane in the analysis of $K_{e4}$ data?
• RQ5 To what extent can systematic biases in form factor extraction be minimized by applying the fitted parameters directly to the initial kinematics?

Key findings

• The upper limit on the branching ratio for $K^+ \to \pi^+\mu^+e^-$ is $<3.9 \times 10^{-11}$, representing a significant improvement over previous limits.
• The $s$-wave $\pi\pi$ scattering length is determined as $a_0^0 = 0.228 \pm 0.012 \pm 0.003$ when the universal band constraint is applied, and $a_0^0 = 0.203 \pm 0.033 \pm 0.004$ without it.
• The form factor $F$ in $K_{e4}$ decay is best described by a quadratic fit with $f_s = 5.75 \pm 0.02 \pm 0.08$ and $f_s' = 1.06 \pm 0.10 \pm 0.40$.
• The form factor $G$ is well described by a linear fit with $g_p = 4.66 \pm 0.47 \pm 0.07$ and $g_p' = 0.67 \pm 0.10 \pm 0.04$.
• The phase shift difference $\delta_{0}^{0} - \delta_{1}^{1}$ is extracted and compared to data, showing good agreement with the Geneva-Saclay results.
• The results are in excellent agreement with updated two-loop chiral perturbation theory predictions, confirming its validity at low energies.