Kaon transverse charge density from space- and timelike data

We used the world data on the kaon form factor to extract the transverse kaon charge density using a dispersion integral of the imaginary part of the kaon form factor in the timelike region. Our analysis includes recent data from $e^{+}{e}^{-}$ annihiliation measurements extending the kinematic reach of the data into the region of high momentum transfers conjugate to the region of short transverse distances. To calculate the transverse density we created a superset of both timelike and spacelike data and developed an empirical parameterization of the kaon form factor. The spacelike set includes two new data points we extracted from existing cross section data. We estimate the uncertainty on the resulting transverse density to be 5% at $b=0.025$ fm and significantly better at large distances. New kaon data planned with the 12 GeV Jefferson Lab may have a significant impact on the charge density at distances of $b<0.1$ fm.

Figure 1

Parameterizations of the charged kaon form factor squared along with world data as a function of $\sqrt{s}$ using (a) parameters from Table 2 in Ref. [41] (dashed black line) and refitted parameters to both spacelike and timelike data using the model in Ref. [41] (dashed-dotted black line). (b) Our best fit (Model 8 in Table 1) with the values listed in Table 2 (solid black line). The error band (dotted gray region) is the 95% confidence band taken from the model error derived from a multinormal distribution of parameters around their means in Table 2. (c) Neutral kaon form factor using the parameterization in Ref. [41] (dashed-dotted black line) along with the refitted parameters to both spacelike and timelike data using the model in Ref. [41] fit to the new high-$s$ data and spacelike data (dashed black line) and our parameterization from Eq. (7) with parameters from Table 2 (solid black line). For (a) and (b), data are taken from [44] ($\mathsf{x}$'s), [45] (open squares), [46] (filled circles), [47] (open circles), [48] (filled squares), [51] (filled triangle; ∼3 GeV), [42] (filled triangle; $\sim 3.67$ GeV), [43] (filled triangles; $\sim 3.7$ and 4.17 GeV), [20] (open triangles), and [24] (filled astroid). For (c), data are from [44] and [52] ($\mathsf{x}$'s), [53] (open squares), and [54] (filled circles). Note that in the timelike region $\sqrt{s}$ takes the place of $\sqrt{t}$.

Figure 2

(a) Comparison of the spacelike dispersion relation of Eq. (8) (short-dashed blue line) and the analytic continuation of our parameterization from Eq. (7) into the spacelike region (solid black line). Data are the same as in Fig. 1. The panels at the right show a dispersion relation calculation (long-dashed red line), the real parts of the Breit-Wigner (BW) functions (solid black line), and the imaginary parts of the BW functions (short-dashed black line) for (b) a single constant-width BW with its dispersion relation calculation (mass, 1020 MeV; width, 4.36 MeV), (c) a single KS BW with an $s$-dependent width and a cutoff of $2m_{K^{+}}$ (mass, 1020 MeV; width, 4.36 MeV), and (d) a single GS BW function with an $s$-dependent width and a cutoff of $2m_{K^{+}}$ (mass, 1020 MeV; width, 4.36 MeV). Note that in the timelike region $\sqrt{s}$ takes the place of $\sqrt{t}$.

Figure 3

(a) Imaginary part of the kaon form factor obtained from our best fit, whose parameters are listed in Table 2 as a function of $\sqrt{t}$ (solid line). The threshold energy is $\sqrt{t}=2m_K$ and indicated by the vertical gray line. The $1\sigma$ confidence bands are included as dashed curves. (b) The weight factor $K_0\left(\sqrt{t}b\right)$ in the dispersion representation of the transverse charge density in Eq. (11) as a function of $\sqrt{t}$ for three values of $b$: 0.1 fm (solid line), 0.5 fm (dashed line), and 2 fm (dashed-dotted line).

Figure 4

Percentage deviation of the dispersion integral of Eq. (11) as a function of the upper-limit integration cutoff. Shown is the contribution to the integral for different transverse distances: $b=0.01$ fm (dashed-dotted line), $b=0.05$ fm (dashed line), and $b=0.1$ fm (solid line). The integrand is evaluated using the parameterization shown in Eq. (7) with parameters from Table 2.

Figure 5

Transverse charge density of the charged kaon using the dispersion integral of Eq. (11) evaluated with the form factor parameterization and fit parameters from Ref. [41] (dashed line), as well as with refitted parameters using the parameterization in Ref. [41] (dashed-dotted line), and our parameterization of Eq. (7) with parameters from Table 2 (solid line). The gray band at the bottom indicates a 95% confidence band of the reparameterization with parameters selected from a multinormal distribution.

Figure 6

Transverse charge density of the charged and neutral kaon (solid and short-dashed lines, respectively) calculated from Eq. (11) compared with the charged pion (dashed-dotted), the proton (long-dashed line), and the neutron (dotted line). Inset: Difference between the charged kaon and the charged pion.

Figure 7

Investigation of the impact of new spacelike data projected for the 12 GeV JLab along with their uncertainties. (a) Values of the projected points calculated in the framework of the Dyson-Schwinger equation (DSE; dashed red line), and perturbative QCD (pQCD; dashed-dotted blue line). Also shown are the monopole curve (dotted black line) and our parameterization with values from Table 2 (solid black line). (b) Impact of the projected data points on our parameterization of the kaon form factor. The dashed red curve uses the values of the form factor calculated in the DSE framework; the dashed-dotted blue curve, those calculated in the pQCD framework. Points on the curve indicate the projected uncertainties of the anticipated measurements. The solid blue curve is Eq. (7) with parameters from Table 2. (c) Impact of the projected points on the transverse charge density of Eq. (11). The difference gives an estimate of the model uncertainty.