Hadron optics in three-dimensional invariant coordinate space from deeply virtual Compton scattering

Abstract

The Fourier transform of the deeply virtual Compton scattering amplitude (DVCS) with respect to the skewness parameter $\ensuremath{\zeta}={Q}^{2}/2p\ifmmode\cdot\else\textperiodcentered\fi{}q$ can be used to provide an image of the target hadron in the boost-invariant variable $\ensuremath{\sigma}$, the coordinate conjugate to light-front time $\ensuremath{\tau}=t+z/c$. As an illustration, we construct a consistent covariant model of the DVCS amplitude and its associated generalized parton distributions using the quantum fluctuations of a fermion state at one loop in QED, thus providing a representation of the light-front wave functions (LFWFs) of a lepton in $\ensuremath{\sigma}$ space. A consistent model for hadronic amplitudes can then be obtained by differentiating the light-front wave functions with respect to the bound-state mass. The resulting DVCS helicity amplitudes are evaluated as a function of $\ensuremath{\sigma}$ and the impact parameter ${\stackrel{\ensuremath{\rightarrow}}{b}}_{\ensuremath{\perp}}$, thus providing a light-front image of the target hadron in a frame-independent three-dimensional light-front coordinate space. Models for the LFWFs of hadrons in $(3+1)$ dimensions displaying confinement at large distances and conformal symmetry at short distances have been obtained using the AdS/CFT method. We also compute the LFWFs in this model in invariant three-dimensional coordinate space. We find that, in the models studied, the Fourier transform of the DVCS amplitudes exhibit diffraction patterns. The results are analogous to the diffractive scattering of a wave in optics where the distribution in $\ensuremath{\sigma}$ measures the physical size of the scattering center in a one-dimensional system.