Parametrization of transition energy-momentum tensor form factors

While the Lorentz structures of the $N \to \Delta$ transition matrix element of the vector current is well-known, that of the symmetric energy-momentum tensor current is unknown. In this letter, we first reproduce the Lorentz structures of the $\frac{1}{2}^{\pm} \to \frac{1}{2}^{\pm}$, $\frac{1}{2}^{\mp} \to \frac{1}{2}^{\pm}$, and $\frac{1}{2}^{\mp} \to \frac{3}{2}^{\pm}$ transition matrix elements of the vector current. We also repeat the parametrizations of the symmetric energy-momentum tensor current for the $\frac{1}{2}^{\pm} \to \frac{1}{2}^{\pm}$ and $\frac{1}{2}^{\mp} \to \frac{1}{2}^{\pm}$ transitions. We then consider both $\frac{1}{2}^{\pm} \to \frac{3}{2}^{\pm}$ and $\frac{1}{2}^{\mp} \to \frac{3}{2}^{\pm}$ matrix elements of the symmetric energy-momentum tensor current (say, $N \to \Delta$ transition) for the first time. We found that there are five independent conserved and four independent non-conserved energy-momentum tensor form factors.