Virtual Compton scattering measurements in the ${\gamma }^{*}N\to \Delta$ transition

We report on new H$(e,e^{\text{'}}p)\gamma$ measurements in the $\Delta (1232)$ resonance at $Q^2=0.06$ (GeV/$c$)${}^2$ carried out simultaneously with H$(e,e^{\text{'}}p){\pi }^0$. It is the lowest $Q^2$ for which the virtual Compton scattering (VCS) reaction has been studied in the first resonance region. The VCS measured cross sections are well described by dispersion-relation calculations in which the multipole amplitudes derived from H$(e,e^{\text{'}}p){\pi }^0$ data are used as input, thus confirming the compatibility of the results. The derived resonant magnetic dipole amplitude $M_{1+}^{3/2}=(40.60\pm 0.{70}_{\mathrm{stat}+\mathrm{sys}})({10}^{-3}/m_{{\pi }^{+}})$ at $W=1232$ MeV is in excellent agreement with the value extracted from H$(e,e^{\text{'}}p){\pi }^0$ measurements.

Figure 1

The derived missing mass spectrum, plotted as a function of the square of missing mass, shows the superb resolution achieved, essential to isolating the small photon decay branch of the $\Delta (1232)$ resonance. Channels for $M_x^2<0.01$ GeV${}^2$ have been multiplied by a factor of ten.

Figure 2

Comparison of the measured VCS cross sections at ${\theta }_{{\gamma }^{*}\gamma }={147}^{°}$ and ${156}^{°}$ as a function of the azimuthal angle ${\varphi }_{{\gamma }^{*}\gamma }$ to Dispersion Relation (DR) calculations [32] with different input. (i) Solid curves, input from the standard solutions of MAID 2003; (ii) long-dashed curves, as in (i) but with $S_{1+}^{3/2}=0$; (iii) dotted curves, as in (i) with $E_{1+}^{3/2}=0$; (iv) dashed-dotted curves, DR results with $S_{1+}^{3/2}=E_{1+}^{3/2}=0$. The shaded band is derived when the dipole and quadrupole amplitudes are constrained to the values extracted from the simultaneously measured H$(e,e^{\text{'}}p){\pi }^0$ [15]. The DR calculation with the parameters ${\Lambda }_{\alpha }=1.0$ GeV and ${\Lambda }_{\beta }=0.6$ GeV (dash-dot-dot curves) is also shown with ${\Lambda }_{\alpha }=0.7$ GeV and ${\Lambda }_{\beta }=0.5$ GeV (solid curves). The uncertainties are the quadratic sum of statistical and systematic errors.

Figure 3

The measured VCS cross sections at ${\theta }_{{\gamma }^{*}\gamma }={180}^{°}$ as a function of $W$. The labeling conventions are the same as those in Fig. 2.

Figure 4

Results for $M_{1+}^{3/2}$ as a function of $Q^2$ from this work (VCS) and from previous pion-electroproduction experiments [3, 6, 12, 15, 24] (errors include statistical and systematic uncertainties). The two results at $Q^2=0.06$ (GeV/$c$)${}^2$ have been shifted by 0.005 (GeV/$c$)${}^2$ to be distinguishable. The theoretical predictions of MAID [21], DMT [19, 20], SAID [22], Sato and Lee [18], and the ChEFT of Pascalutsa and Vanderhaeghen [34] are also shown.