Scaling Tests of the Cross Section for Deeply Virtual Compton Scattering

We present the first measurements of the $\overrightarrow{e}p\to ep\gamma$ cross section in the deeply virtual Compton scattering (DVCS) regime and the valence quark region. The $Q^2$ dependence (from 1.5 to $2.3{\mathrm{GeV}}^2$) of the helicity-dependent cross section indicates the twist-2 dominance of DVCS, proving that generalized parton distributions (GPDs) are accessible to experiment at moderate $Q^2$. The helicity-independent cross section is also measured at $Q^2=2.3{\mathrm{GeV}}^2$. We present the first model-independent measurement of linear combinations of GPDs and GPD integrals up to the twist-3 approximation.

Figure 1

Lowest-order QED diagrams for the process $ep\to ep\gamma$, including the DVCS and Bethe-Heitler (BH) amplitudes. The external momentum four-vectors are defined on the diagram. The virtual photon momenta are $q=k-k^{\prime }$ in the DVCS- and $\Delta =q-q^{\prime }$ in the BH-amplitudes. The invariants are: $W^2=(q+p{)}^2$, $Q^2=-q^2>0$, $t={\Delta }^2$, $x_{\mathrm{Bj}}=Q^2/(2p·q)$, and the DVCS scaling variable $\xi =-{\overline{q}}^2/(\overline{q}·P)\approx x_{\mathrm{Bj}}/(2-x_{\mathrm{Bj}})$, with $\overline{q}=(q+q^{\prime })/2$ and $P=p+p^{\prime }$.

Figure 2

Missing mass squared for $H(e,e^{\prime }\gamma )X$ events (stars) at $Q^2=2.3{\mathrm{GeV}}^2$ and $-t\in [0.12,0.4]{\mathrm{GeV}}^2$, integrated over the azimuthal angle of the photon ${\varphi }_{\gamma \gamma }$. The solid histogram shows the data once the $H(e,e^{\prime }\gamma )\gamma X^{\prime }$ events have been subtracted. The other histograms are described in the text.

Figure 3

Data and fit to $d^4\Sigma /[d{Q}^{2}d{x}_{\mathrm{Bj}}dtd{\varphi }_{\gamma \gamma }]$, and $d^4\sigma /[d{Q}^{2}d{x}_{\mathrm{Bj}}dtd{\varphi }_{\gamma \gamma }]$, as a function of ${\varphi }_{\gamma \gamma }$. Both are in the bin $⟨Q^2,t⟩=(2.3,-0.28){\mathrm{GeV}}^2$ at $⟨x_{\mathrm{Bj}}⟩=0.36$. Error bars show statistical uncertainties. Solid lines show total fits with one-$\sigma$ statistical error bands. Systematic uncertainty is given in the text. The dot-dot-dashed line is the $|\mathrm{BH}{|}^2$ contribution to $d^4\sigma$. The short-dashed lines in $d^4\Sigma$ and $d^4\sigma$ are the fitted $\mathrm{Im}$ and $\mathrm{Re}$ parts of ${\mathcal{C}}^{\mathcal{I}}(\mathcal{F})$, respectively. The long-dashed line is the fitted $\mathrm{Re}[{\mathcal{C}}^{\mathcal{I}}+\Delta {\mathcal{C}}^{\mathcal{I}}](\mathcal{F})$ term. The dot-dashed curves are the fitted $\mathrm{Im}$ and $\mathrm{Re}$ parts of ${\mathcal{C}}^{\mathcal{I}}({\mathcal{F}}^{\mathrm{eff}})$.

Figure 4

Left: $Q^2$ dependence of $\mathrm{Im}$ parts of (twist-2) ${\mathcal{C}}^{\mathcal{I}}(\mathcal{F})$ and (twist-3) ${\mathcal{C}}^{\mathcal{I}}({\mathcal{F}}^{\mathrm{eff}})$ angular harmonics, averaged over $t$. The horizontal line is the fitted average of $\mathrm{Im}[{\mathcal{C}}^{\mathcal{I}}(\mathcal{F})]$. Right: Extracted real and imaginary parts of the twist-2 angular harmonics as functions of $t$. The VGG model curves are described in the text. Note the sign of $-[{\mathcal{C}}^{\mathcal{I}}+\Delta {\mathcal{C}}^{\mathcal{I}}](\mathcal{F})$ (data and VGG). Superposed points in both panels are offset for visual clarity. Their error bars show statistical uncertainties.