Exposing strangeness: Projections for kaon electromagnetic form factors

Fei Gao, Lei Chang, Yu-Xin Liu, Craig D. Roberts, and Peter C. Tandy

Phys. Rev. D 96, 034024 – Published 28 August 2017

Abstract

A continuum approach to the kaon and pion bound-state problems is used to reveal their electromagnetic structure. For both systems, when used with parton distribution amplitudes appropriate to the scale of the experiment, Standard Model hard-scattering formulas are accurate to within 25% at momentum transfers $Q^{2} \approx 8 {GeV}^{2}$ . There are measurable differences between the distribution of strange and normal matter within the kaons, e.g. the ratio of their separate contributions reaches a peak value of 1.5 at $Q^{2} \approx 6 {GeV}^{2}$ . Its subsequent $Q^{2}$ evolution is accurately described by the hard scattering formulas. Projections for the ratio of kaon and pion form factors at timelike momenta beyond the resonance region are also presented. These results and projections should prove useful in planning next-generation experiments.

Received 14 March 2017

DOI:https://doi.org/10.1103/PhysRevD.96.034024

Physics Subject Headings (PhySH)

Form factors Quantum chromodynamics

Kaons Pions

Particles & Fields

Authors & Affiliations

Fei Gao^1,2,*, Lei Chang^3,†, Yu-Xin Liu^1,2,4,‡, Craig D. Roberts^5,§, and Peter C. Tandy^6,∥

¹Department of Physics and State Key Laboratory of Nuclear Physics and Technology, Peking University, Beijing 100871, China
²Collaborative Innovation Center of Quantum Matter, Beijing 100871, China
³School of Physics, Nankai University, Tianjin 300071, China
⁴Center for High Energy Physics, Peking University, Beijing 100871, China
⁵Physics Division, Argonne National Laboratory, Argonne, Illinois 60439, USA
⁶Center for Nuclear Research, Department of Physics, Kent State University, Kent, Ohio 44242, USA

^*hiei@pku.edu.cn
^†leichang@nankai.edu.cn
^‡yxliu@pku.edu.cn
^§cdroberts@anl.gov
^∥tandy@kent.edu

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Vol. 96, Iss. 3 — 1 August 2017

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Images

Figure 1
Solid curve—our $K^{+}$ form factor; and dot-dashed curve (green)—result from Ref. [15]. Data: Refs. [9, 10].
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Figure 2
Upper panel: Solid curve, Eq. (3) prediction for $Q^{2} F_{K} (Q^{2})$ ; dot-dashed curve (indigo), result from Ref. [15], which is limited to the domain $Q^{2} < 4 {GeV}^{2}$ ; and dashed curve and band (green), result produced by the hard scattering formula, Eqs. (1), (5). Filled diamonds, data anticipated from a forthcoming experiment [28]: The two error estimates differ in their assumptions about the $t$ and model dependence of the form-factor extractions [29]. Lower panel: Solid curve: Prediction for $Q^{2} F_{π} (Q^{2})$ ; and dashed curve (blue), result produced by the hard scattering formula, Eqs. (1), (4). Data: Star [23], circles and squares [50]; and diamonds and triangle, anticipated reach and accuracy of forthcoming experiments [25, 26]. In both panels, the dotted curve (red) is Eq. (1) computed with the conformal-limit PDA, $φ^{cl} (x) = 6 x (1 - x)$ .
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Figure 3
Solid curve, ratio of $\bar{s}$ - and $u$ -quark contributions to the $K^{+}$ form factor; and dashed curve and band (green), prediction of the hard-scattering formula, Eqs. (1), (5).
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Figure 4
Flavor separation of kaon form factor and comparison with complete $π$ and $K$ results. Left panel: Solid (blue) curve, $u_{K} = F_{K}^{u} (Q^{2})$ , i.e. $u$ -quark in the $K^{+}$ ; dot-dashed (green) curve, $s_{K} = F_{K}^{\bar{s}}$ ; dashed (black) curve, $u_{π} = F_{π}$ (we assume isospin symmetry); and dotted (red) curve, $F_{K}$ . Right panel: $Q^{2}$ -weighted form factors. The (blue) band containing the solid (blue) curve is the hard-scattering formula prediction for $Q^{2} u_{K}$ , and the (green) band and curve correspond to $Q^{2} s_{K}$ , both obtained using Eq. (5) in Eq. (2) for the appropriate flavour and including the multiplicative factors specified in Eq. (1). Recall that quark-charge factors are not included when defining the quark contributions, Eq. (3a), so all form factors are unity at $Q^{2} = 0$ .
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Figure 5
Upper panel: Charged kaon-to-pion ratio. Solid curve at spacelike momenta, direct calculation, Eq. (3). Dashed curve and band (green), prediction for the timelike behavior, derived from Eqs. (1), (4), (5). Data from Ref. [30]. Lower panel: Ratio of neutral-to-charged kaon form factors. Solid curve, direct calculation. We compute $r_{K^{0}}^{2} = - (0.21 fm)^{2}$ cf. experiment [66]: $r_{K^{0}}^{2} = - (0.24 \pm 0.08 fm)^{2}$ . Dashed curve and band (green), prediction for the timelike behavior of this ratio derived from Eqs. (1), (5). Datum from Ref. [31]: the error bar marks the 90% confidence interval.
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