Pionic transparency in semi-exclusive electroproduction off nuclei

We investigate the early onset of pionic color transparency (πCT) observed at Jefferson Laboratory (JLAB) in semi-exclusive pion electroproduction reaction $A(e,e^{\text{'}}{\pi }^{+})$ off nuclei. In the present description the primary ${\gamma }^{*}p\to {\pi }^{+}n$ interaction is described very well both for the longitudinal and the transverse polarizations. For the final state interactions a coupled-channel treatment of the interactions of transmitted hadrons allows to go beyond the Glauber approximation. We show that a proper distinction between the soft hadronic and hard partonic components of the electroproduction amplitude is essential for a quantitative description of the measured nuclear transparency. The data are well reproduced if one assumes that point-like configurations are produced in the regime of hard deep-inelastic scattering (DIS) off partons and dominate the transverse channel.

Figure 1

The diagrams describing the hadron–exchange part of the ${\pi }^{+}\text{−}$electroproduction amplitude. See text for the details.

Figure 2

A schematic representation of the partonic DIS part of the ${\pi }^{+}\text{−}$electroproduction mechanism. The wavy line represents a color string. See text for the details.

Figure 3

The longitudinal $d{\sigma }_{\mathrm{L}}/\mathit{dt}$ (top panels) and transverse $d{\sigma }_{\mathrm{T}}/\mathit{dt}$ (bottom panels) differential cross sections of the reaction $p(e,e^{\text{'}}{\pi }^{+})n$ at average values of $Q^2=2.15\hspace{0.5em}{\mathrm{GeV}}^2$ and $Q^2=3.91\hspace{0.5em}{\mathrm{GeV}}^2$. The solid curves are the contribution of the hadron-exchange model and the histograms are the contribution of the DIS pions. The πCT data are from Ref. [36]. The discontinuities in the curves result from the different values of $Q^2$ and $W$ for the various $-t$ bins.

Figure 4

Left panel: Decomposition of the the integrated virtual photon-nucleon cross sections of the exclusive reaction $p(e,e^{\text{'}}{\pi }^{+})n$ as a function of the photon virtuality $Q^2$ at $W=2.2$ GeV. The contribution of the hadron-exchange model to the transverse ${\sigma }_{\mathrm{T}}$ and longitudinal ${\sigma }_{\mathrm{L}}$ cross sections are shown by solid and dashed curves, respectively. The contribution of DIS pions to ${\sigma }_{\mathrm{T}}$ is shown by the dash-dotted curve. Right panel: The forward ${\pi }^{+}$ differential cross sections as a function of $Q^2$.

Figure 5

The Lund model average formation time of (pre)pions (solid curve) in the target rest frame as a function of $Q^2$ in the forward kinematics of the πCT experiment. The dashed curve is an estimate based on Eq. (3) with $\Delta M^2=1\hspace{0.5em}{\mathrm{GeV}}^2$.

Figure 6

Transparency, $T_A$, vs. $Q^2$ for ${}^{12}\mathrm{C}$ (left, top panel), ${}^{27}\mathrm{Al}$ (right, top), ${}^{63}\mathrm{Cu}$ (left, bottom) and ${}^{197}\mathrm{Au}$ (right, bottom). The dotted curves correspond to FSI with the full hadronic cross section and the dashed curves include the shadowing corrections. The dash-dotted curves correspond to the in-medium cross sections defined according to the Lund model formation time concept which includes the $Q^2\text{−}$dependent (pre)hadronic interactions, Eq. (5), for the transverse contribution. The solid curves describe the effect of time dilatation alone with the pedestal value in the effective cross section independent of $Q^2$. The dash-dash-dotted curve in the top left panel realises the CT effect both in the longitudinal and transverse channels. The experimental data are from Ref. [20].

Figure 7

Transparency, $T_A$, vs. $Q^2$ for ${}^{12}\mathrm{C}$ (left, top panel), ${}^{27}\mathrm{Al}$ (right, top), ${}^{63}\mathrm{Cu}$ (left, bottom) and ${}^{197}\mathrm{Au}$ (right, bottom). The formation time of (pre)pions in the laboratory is calculated using Eq. (3). The dash-dash-dotted curves realize the CT effect in both the longitudinal and transverse channels and dash-dotted curves in the transverse channel only. The dot-dot-dashed curves describe the CT effect in the longitudinal channel only. The experimental data are from Ref. [20].