Constraining scenarios of the soft/hard transition for the pion electromagnetic form factor with expected data of 12 GeV Jefferson Lab experiments and of the electron-ion collider

It has been shown previously [S. V. Troitsky and V. E. Troitsky, Phys. Rev. D 88, 093005 (2013)] that a nonperturbative relativistic constituent-quark model for the $\pi$-meson electromagnetic form factor allows for a quantitative description of the soft/hard transition, resulting in the correct quantum-chromodynamical asymptotics, including normalization, from the low-energy data without further parameter tuning. This happens universally whenever the constituent-quark mass is switched off. The energy range where the transition happens is therefore determined by the quark-mass running at intermediate energies and is not tightly constrained theoretically. Here we consider possible ways to pin down this energy range with coming experimental data. We demonstrate that expected experimental uncertainties of the 12 GeV Jefferson Lab data are larger than the span of predictions of the model, so these data might be used for testing the model but not for determination of the soft/hard transition scale. Contrarily, the projected electron-ion collider will be capable of pinning down the scale.

Figure 1

Allowed regions of the plane of parameters $\mu$ and $\lambda$ which govern the quark-mass evolution; see the text. Long-dashed line: the consistency limit (the region to the left of the line is allowed; relevant for all cases). Full black line: the 95% C.L. bound from the present data (the region to the right of the line is allowed). Other lines: 95% C.L. example bounds from future data for the low-scale (dashed; the allowed region is bound by the lines) and high-scale (dot-dashed; the allowed region is to the right from the lines) scenarios of the soft/hard transition; thin lines assume 12 GeV Jefferson Laboratory data, and thick gray lines assume EIC data. The values of $\mu$ and $\lambda$ assumed for the low-scale and high-scale scenarios are shown by the triangle and the diamond, respectively.

Figure 2

The range of predictions of our model for the charged pion form factor allowed by the present data and model consistency for various scenarios of quark-mass running (gray band) together with predictions of Refs. [22] (area between two dotted lines) and [23] (dash-dotted line). Two dashed lines represent two particular scenarios for $M(Q^2)$ corresponding to the low-scale and high-scale soft/hard transitions discussed in the text. Existing experimental data points (see Ref. [4] for their description and list of references) are shown by black dots with thin error bars. A typical simulated example realization of expected 12 GeV Jefferson Laboratory data, corresponding to the high-scale scenario, is shown by gray dots with thick gray error bars. It is evident that the Jefferson Laboratory data would not help to choose between the allowed scenarios of quark-mass running within our model but would provide a good test of the model vs others.

Figure 3

The range of predictions of our model for the charged pion form factor allowed by simulated EIC data and model consistency for two particular scenarios for $M(Q^2)$ (dashed lines) corresponding to the low-scale and high-scale soft/hard transitions discussed in the text (gray bands bound by dotted lines). Existing experimental data points (see Ref. [4] for their description and list of references) are shown by black dots with thin error bars. Two typical simulated example realizations of expected EIC data, corresponding to the high-scale (black) and low-scale (gray) scenarios, are shown by dots with thick error bars. JLab 12-GeV simulated data are not shown for clarity. It is evident that the EIC data would make it possible to choose between the allowed scenarios of quark-mass running within our model.