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Resonant elastic X-ray scattering is a powerful technique for measuringmultipolar order parameters. In this paper, we theoretically and experimentallystudy the possibility of using this technique to detect the proposed multipolarorder parameters in URu$_2$Si$_2$ at the U-$L_{3}$ edge with the electricquadrupolar transition. Based on an atomic model, we calculate the azimuthaldependence of the quadrupolar transition at the U-$L_{3}$ edge. The resultsillustrate the potential of this technique for distinguishing differentmultipolar order parameters. We then perform experiments on ultra-clean singlecrystals of URu$_2$Si$_2$ at the U-$L_{3}$ edge to search for the predictedsignal, but do not detect any indications of multipolar moments within theexperimental uncertainty. We theoretically estimate the orders of magnitude ofthe cross-section and the expected count rate of the quadrupolar transition andcompare them to the dipolar transitions at the U-$M_4$ and U-$L_3$ edges,clarifying the difficulty in detecting higher order multipolar order parametersin URu$_2$Si$_2$ in the current experimental setup.

On the possibility to detect multipolar order in URu$_2$Si$_2$ by the electric quadrupolar transition of resonant elastic X-ray scattering