Pion-kaon scattering amplitude constrained with forward dispersion relations up to 1.6 GeV

In this work we provide simple and precise parametrizations of the existing $\pi K$ scattering data from threshold up to 1.6 GeV, which are constrained to satisfy forward dispersion relations as well as three additional threshold sum rules. We also provide phenomenological values of the threshold parameters and of the resonance poles that appear in elastic scattering.

Figure 1

Experimental data on the $S^{3/2}$ phase shift, ${\delta }_0^{3/2}(s)$. The data come from [18] (Cho et al.), [19] (Bakker et al.), [20] (Jongejans et al.), [21] (Linglin et al.) and [22] (Estabrooks et al.). The dashed line shows our fit to these data and the dotted lines enclose its uncertainty band. The continuous line represents our unconstrained fit including also the data on $t_0^{1/2}+t_0^{3/2}/2$, whose uncertainty is represented by the gray band.

Figure 2

Data on ${\widehat{t}}_S(s)$ from Estabrooks et al. [22] and Aston et al. [24]. The upper panel shows $|{\widehat{t}}_S(s)|$, whereas the lower one shows ${\varphi }_S(s)$, which were measured independently. The continuous line is our unconstrained fit (UFD), whose uncertainties are covered by the gray band. For comparison we show, as a dashed line, a fit only to the data in [24], whose corresponding uncertainties are delimited by the dotted lines.

Figure 3

$S^{1/2}$-wave phase shift below the $K\eta$ threshold. In this region the amplitude is elastic in practice. The continuous line is our UFD result, whose uncertainty is covered by the gray band. Data points extracted from Estabrooks et al. [22] and Aston et al. [24]. As explained in the text, we do not fit this wave individually, but in combination with the $I=3/2$, as it was originally measured. The dashed curve is the fit to Aston et al. [24] data alone and the dotted lines cover the corresponding uncertainty band.

Figure 4

Data on the $I=3/2$ $P$ wave from Estabrooks et al. [22]. We also show our UFD result as solid line with a gray uncertainty band, which is obtained by fitting these data together with the data on the $t_P=t_1^{1/2}+t_1^{3/2}/2$ combination. For comparison we show with a dashed line a fit only to the data in this figure, whose uncertainty is delimited by the dotted lines.

Figure 5

Data on $|{\widehat{t}}_P(s)|$ from [22, 24]. The continuous line is our unconstrained fit (UFD), whose uncertainties are covered by the gray band. For comparison we show as a dashed line a fit only to the data from [24], whose corresponding uncertainties are delimited by the dotted lines.

Figure 6

Data on ${\varphi }_P(s)$ from Estabrooks et al. [22] and Aston et al. [24]. The continuous line is our unconstrained fit (UFD), whose uncertainties are covered by the gray band. For comparison we show as a dashed line a fit only to the data from [24], whose corresponding uncertainties are delimited by the dotted lines.

Figure 7

$P^{1/2}$-wave phase shift below the $K\eta$ threshold. The continuous line is our UFD parametrization, whose uncertainty is covered by the gray band. Data from Estabrooks et al. [22] and Aston et al. [24]. The dashed curve is the fit to Aston et al. data alone and the dotted lines cover its corresponding uncertainty.

Figure 8

Data on the $I=3/2$ $D$ wave from Estabrooks et al. [22]. We also show our UFD result as a solid line with a gray uncertainty band, which is obtained by fitting this data in a simultaneous fit with the data on the $t_2^{1/2}+t_2^{3/2}/2$ combination. For comparison we show with a dashed line a fit only to the data in this figure.

Figure 9

Data on $|{\widehat{t}}_D(s)|$ from [22, 24]. The continuous line is our unconstrained fit (UFD), whose uncertainties are covered by the gray band. For comparison we show, as a dashed line, a fit only to the data from [24], whose corresponding uncertainties are delimited by the dotted lines.

Figure 10

Data on ${\varphi }_D(s)$ from [22, 24]. The continuous line is our unconstrained fit (UFD), whose uncertainties are covered by the gray band. For comparison we show, as a dashed line, a fit only to the data from [24], whose corresponding uncertainties are delimited by the dotted lines.

Figure 11

Data on $|{\widehat{t}}_F(s)|$ from [22, 24]. The continuous line is our unconstrained fit (UFD), whose uncertainties are covered by the gray band. For comparison we show, as a dashed line, a fit to the data from [24], whose corresponding uncertainties are delimited by the dotted lines.

Figure 12

Data on ${\varphi }_F(s)$ from [22, 24]. The continuous line is our unconstrained fit (UFD), whose uncertainties are covered by the gray band. For comparison we show, as a dashed line, a fit to the data from [24], whose corresponding uncertainties are delimited by the dotted lines.

Figure 13

Comparison between the input (dashed lines) and the output for the $T^{+}$ (top panel) and $T^{-}$ (bottom panel) FDRs when using the UFD set. These correspond to the left-hand side versus the right-hand side in Eqs. (49) and (50). The gray bands describe the uncertainty of the difference.

Figure 14

Comparison between the input (dashed line) and the output for the $T^{+}$ (top panel) and $T^{-}$ (bottom panel) FDRs when using the CFD set. These correspond to the left-hand side versus the right-hand side, respectively, in Eqs. (49) and (50). The gray bands describe the uncertainty of the difference. Note the dramatic improvement below 1.6 GeV compared to the results in Fig. 13 using the UFD set. Actually, the input and dispersive CFD calculations are consistent within uncertainties up to 1.6 GeV.

Figure 15

The CFD parametrization of the $S^{3/2}$ wave is shown as a continuous line and its uncertainty as a gray band. For comparison the UFD parametrization is shown as a dashed line. See Fig. 1 for data references.

Figure 16

CFD parametrization of the $S^{1/2}$-wave phase shift in the elastic region, shown as a continuous line whose uncertainty is covered by the gray band. For comparison we also show as a dashed line the UFD result. Note that in this case the UFD and CFD parametrizations are almost indistinguishable. See Fig. 3 for data references.

Figure 17

CFD parametrization of the ${\widehat{t}}_S$ wave in the whole energy region. On the upper panel we show $|{\widehat{t}}_S(s)|$, whereas in the lower one we show ${\varphi }_S(s)$. The continuous line is our constrained fit (CFD), whose uncertainties are covered by the gray band. For comparison we show the UFD result as a dashed line. Data references are as in Fig. 2.

Figure 18

Phase shift of the $P^{3/2}$ wave. Data are from [22]. We show as a continuous line the CFD fit and the gray band covers its uncertainties. The UFD result lies right on the border of this uncertainty band. Note that this phase is rather small up to 1.74 GeV.

Figure 19

Phase shift of the $P^{1/2}$ wave. The CFD and UFD results are almost indistinguishable up to 950 MeV, where the CFD phase becomes somewhat smaller. Note however that the UFD results still lie inside the uncertainty band. In addition we show the solution in [14]. Data are from [22, 24].

Figure 20

Modulus of the ${\widehat{t}}_P={\widehat{t}}_1^{1/2}+{\widehat{t}}_1^{3/2}/2$ amplitude. We show the CFD fit as a continuous line and its uncertainty as a gray band. Note that the UFD result (dashed line) is also compatible within the CFD uncertainties. Data are from [22, 24].

Figure 21

Phase of the $t_P=t_1^{1/2}+t_1^{3/2}/2$ amplitude. We show the CFD fit as a continuous line and its uncertainty as a gray band. Note that the UFD result (the dashed line) is also very compatible within the CFD uncertainties. Data are from [22, 24].

Figure 22

$D^{3/2}$ phase shift. Data are from [22]. We show the CFD result as a continuous line and its uncertainty as a gray band. Note that it deviates very slightly from the UFD result (the dashed line) and only above 1.2 GeV.

Figure 23

Modulus of the ${\widehat{t}}_D$ wave. Data are from [22, 24]. Note the clear peak of the $K_2^{*}(1430)$ resonance and that above 1.6 GeV the UFD central result (the dashed line) is incompatible with the CFD result (the continuous line) within its uncertainties (the gray band).

Figure 24

Phase of the $t_D$ wave. Data are from [22, 24]. Note the sharp phase rise due to the $K_2^{*}(1430)$ resonance and that above 1.6 GeV the existing data and the UFD central result (the dashed line) are both incompatible with the CFD result (the continuous line) within its uncertainties (the gray band).

Figure 25

Modulus of the $F^{1/2}$ wave. Data are from [22, 24]. Note that the peak of the $K_3^{*}(1780)$ resonance is well described by both the CFD and UFD curves, which are very compatible. The gray band stands for the CFD uncertainty.

Figure 26

Phase of the $F^{1/2}$ wave. Data are from [22, 24]. Note that the CFD and UFD curves are very compatible. The gray band stands for the CFD uncertainty.

Figure 27

Phase of the $I=1/2$ $\pi K$ $S$ wave obtained from the semileptonic decay $D^{+}\to K^{-}{\pi }^{+}{e}^{+}{\nu }_e$ by the BABAR Collaboration [25] and recently by the BESIII Collaboration [26]. These phases are compared to the LASS scattering phase shift of Aston et al. (using their $I=3/2$ parametrization to separate the $I=1/2$). Note that the experiments are in fairly good agreement up to 1.6 GeV.

Figure 28

Phase of the $I=1/2$ $\pi K$ $S$ wave obtained from Dalitz plot analyses of $D\to K\pi \pi$ by the E791 Collaboration [27] and of ${\eta }_c\to K_S^0{K}^{\pm }{\pi }^{\mp }$ by the BABAR Collaboration [30]. We have plotted the systematic plus the statistical uncertainties for [27]. These phases are compared to the LASS scattering phase shift of Aston et al. (using their $I=3/2$ parametrization to separate the $I=1/2$). The data from BABAR are displaced by 34° while those from E791 are displaced by 86°. Note that the qualitative agreement with the scattering phase only reaches up to 1.5 GeV and is not particularly good at low energies.

Figure 29

Analytic structure in different variables of a $\pi K$ scattering partial wave $t(s)$ and effective range function $\mathrm{\Phi }(s)$: (a) $t(s)$ in the complex $s$ plane. Note the elastic, inelastic, left-hand and circular cuts. (b) $\mathrm{\Phi }(s)$ in the $s$ plane has the same structure as $t(s)$ except for the absence of the elastic cut. (c) In the $y(s)$ plane the circular cut disappears. (d) The conformal variable $\omega (y)$ maps the whole analyticity domain of $\mathrm{\Phi }(y)$ inside the unit circle, whereas the cut singularities are confined to $|\omega |=1$. Note that $\omega$ is defined so that the data region is roughly centered around $\omega =0$ and not too close to the border.