Complete definition of $N\to \mathrm{\Delta }$ transition generalized parton distributions

We revisit the definition of the leading-twist chiral-even generalized parton distributions (GPDs) for $N\to \mathrm{\Delta }$ baryon transitions. We identify and address deficiencies in previous definitions of the transition GPDs inspired by the transition form factors of the vector and axial-vector currents. Through systematic analysis of all possible covariant structures, respecting discrete symmetries and the baryon spinor equations of motion, we derive complete sets of independent structures for the transition matrix elements of the vector and axial-vector partonic operators. They contain additional structures proportional to the light-cone vector, corresponding to transition GPDs of vanishing first moment, which were not included in previous parametrizations. Their presence is confirmed independently by the light-front multipole expansion and the cross-channel SO(3) partial-wave analysis of the transition matrix elements. Our analysis provides a complete definition of the $N\to \mathrm{\Delta }$ transition GPDs for use in theoretical and phenomenological studies.