Measuring the phase of the scattering amplitude with vortex beams

We show that colliding vortex beams instead of (approximate) plane waves can lead to a direct measurement of how the overall phase of the plane wave scattering amplitude changes with the scattering angle. Since vortex beams are coherent superpositions of plane waves with different momenta, their scattering amplitude receives contributions from plane wave amplitudes with distinct kinematics. These contributions interfere, leading to the measurement of their phase difference. Although interference exists for any generic wave packet collision, we show that using vortex beams dramatically enhances sensitivity to the phase in comparison with nonvortex beams. Since the overall phase is inaccessible in a plane wave collision, this measurement would be of great importance for a number of topics in hadronic physics, for example, meson production in the resonance region, physics of nucleon resonances, and small angle elastic hadron scattering.

Figure 1

Two interfering kinematical configurations on the transverse plane for vectors ${\mathbf{k}}_1$ and ${\mathbf{k}}_2$ of fixed length that sum up to the same vector $\mathbf{K}$.

Figure 2

Quantity $|\mathbf{K}|R_{\perp }$ as a function of $|\mathbf{K}|$ (measured in arbitrary units) for $m_1=0$ (left) and $m_1=3$ (right); $m_2=0$ in both cases. The plots were drawn for ${\varkappa }_1=1$ and ${\varkappa }_2=0.5$ measured in the same arbitrary units. The solid and dashed lines correspond to ${\sigma }_i={\varkappa }_{0i}/50$ and ${\sigma }_i={\varkappa }_{0i}/5$, respectively.

Figure 3

Schematic representation of the pion photoproduction kinematics in the proton “rest frame” in the case when both the photon and the initial proton are represented by the Bessel vortex states $|{\varkappa }_{\gamma },m_{\gamma }⟩$ and $|{\varkappa }_p,m_p⟩$.

Figure 4

Detecting noncollinearity of the two final transverse momenta is a prerequisite to measuring the azimuthal asymmetry.

Figure 5

The ratio $\mathcal{P}$ defined in (41) for the transverse Bessel wave packets as a function of photon’s orbital helicity $m_{\gamma }$; the proton’s orbital helicity is fixed at $m_p=0$. The calculations are done for ${\varkappa }_{\gamma }=2{\varkappa }_p$; the solid and dashed lines correspond to ${\sigma }_i={\varkappa }_{0i}/5$ and ${\sigma }_i={\varkappa }_{0i}/50$, respectively.