[Paper Review] The Kpi vector form factor and constraints from Kl3 decays

Diogo Boito, Rafel Escribano+1 more|arXiv (Cornell University)|Jan 14, 2011

Particle physics theoretical and experimental studies11 references3 citations

TL;DR

This paper presents a dispersive representation of the $K\pi$ vector form factor using three subtractions and two resonances, fitting $\tau\to K\pi\nu_{\tau}$ and $K_{l3}$ decay data to extract the slope $\lambda_{+}^{\prime} = (25.49 \pm 0.31) \times 10^{-3}$ and curvature $\lambda_{+}^{\prime\prime} = (12.22 \pm 0.14) \times 10^{-4}$, with a $K^*(892)^\pm$ pole at $m = 892.0 \pm 0.5$ MeV and $\Gamma = 46.5 \pm 1.1$ MeV, improving precision over previous averages by combining multiple experimental inputs.

ABSTRACT

The slope and curvature parameters of the $Kπ$ vector form factor, $F_+^{Kπ}$, are fitted to the data on $ auKpi$ and $K_{l3}$ decays yielding $λ_+'=(25.49 \pm 0.31) imes 10^{-3}$ and $λ_+"= (12.22 \pm 0.14) imes 10^{-4}$. The pole position of the $K^*(892)^\pm$ is found to be at $m_{K^*(892)^\pm}= 892.0\pm 0.5$ MeV and $Γ_{K^*(892)^\pm}= 46.5 \pm1.1$ MeV. The phase-space integrals relevant for $K_{l3}$ analyses and the $P$-wave isospin-1/2 $Kπ$ phase-shift threshold parameters are also calculated.

Motivation & Objective

• To determine the $K\pi$ vector form factor $F_+^{K\pi}(q^2)$ with high precision using combined data from $K_{l3}$ and $\tau\to K\pi\nu_{\tau}$ decays.
• To extract the slope $\lambda_{+}^{\prime}$ and curvature $\lambda_{+}^{\prime\prime}$ parameters of the form factor's Taylor expansion around $q^2=0$.
• To determine the pole position of the $K^*(892)^\pm$ resonance from the form factor's analytic structure.
• To calculate phase-space integrals relevant for $K_{l3}$ decay analyses and $P$-wave $K\pi$ isospin-1/2 scattering threshold parameters.

Proposed method

• A dispersive representation of the normalized vector form factor $\tilde{F}_+(s)$ is employed with three subtractions and two resonances to ensure analyticity and unitarity.
• The form factor is parameterized as $\tilde{F}_+(s) = \exp\left[\alpha_1 \frac{s}{m_{\pi}^2} + \frac{1}{2}\alpha_2 \frac{s^2}{m_{\pi}^4} + \frac{s^3}{\pi} \int_{s_{K\pi}}^{s_{\rm cut}} \frac{\delta(s')}{s'^3(s'-s-i0)} ds' \right]$, where $\delta(s)$ is the phase of the form factor.
• The subtraction constants $\alpha_1 = \lambda_{+}^{\prime}$ and $\alpha_2 = \lambda_{+}^{\prime\prime} - \lambda_{+}^{\prime 2}$ are determined by fitting to experimental data from $K_{l3}$ and $\tau\to K\pi\nu_{\tau}$ decays.
• The $K^*(892)^\pm$ pole position is extracted from the complex $s$-plane singularity of the form factor, defined via $\sqrt{s_{K^*}} = m_{K^*} - i\Gamma_{K^*}/2$.
• Phase-space integrals for $K_{l3}$ decays are computed numerically using the fitted form factor.
• Threshold parameters for $P$-wave $K\pi$ scattering are extracted from the Taylor expansion of the $T$-matrix near threshold using the phase from the form factor.

Experimental results

Research questions

• RQ1 What are the precise values of the slope $\lambda_{+}^{\prime}$ and curvature $\lambda_{+}^{\prime\prime}$ of the $K\pi$ vector form factor in the $K_{l3}$ decay region?
• RQ2 What is the pole position of the $K^*(892)^\pm$ resonance as determined from the full analytic structure of the form factor?
• RQ3 How do the phase-space integrals for $K_{l3}$ decays depend on the form factor, and what are their updated values?
• RQ4 What are the $P$-wave $K\pi$ isospin-1/2 scattering threshold parameters derived from the $\tau\to K\pi\nu_{\tau}$ decay spectrum?
• RQ5 How do the results compare with previous averages, particularly in light of the high-precision Belle and BaBar data?

Key findings

• The slope parameter is determined as $\lambda_{+}^{\prime} \times 10^3 = 25.49 \pm 0.31$, with a significant improvement in precision over previous averages.
• The curvature parameter is found to be $\lambda_{+}^{\prime\prime} \times 10^4 = 12.22 \pm 0.14$, with reduced uncertainties due to combined $K_{l3}$ and $\tau$ decay constraints.
• The $K^*(892)^\pm$ resonance mass is measured at $892.0 \pm 0.5$ MeV and width at $46.5 \pm 1.1$ MeV, consistent with the PDG values but with improved precision.
• Phase-space integrals for $K_{l3}$ decays are computed as $I_{K^{0}_{e3}} = 0.15466(17)$, $I_{K^{0}_{\mu 3}} = 0.10276(10)$, $I_{K^{+}_{e3}} = 0.15903(17)$, and $I_{K^{+}_{\mu 3}} = 0.10575(11)$, with uncertainties properly propagated.
• The $P$-wave $K\pi$ scattering threshold parameters are extracted as $m_{\pi}^3 a_1^{1/2} \times 10 = 0.166(4)$, $m_{\pi}^5 b_1^{1/2} \times 10^2 = 0.258(9)$, and $m_{\pi}^7 c_1^{1/2} \times 10^3 = 0.90(3)$, providing new constraints on $K\pi$ interactions.