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We present a model-independent approach to electric quadrupole transitions ofdeformed nuclei. Based on an effective theory for axially symmetric systems,the leading interactions with electromagnetic fields enter as minimal couplingsto gauge potentials, while subleading corrections employ gauge-invariantnon-minimal couplings. This approach yields transition operators that areconsistent with the Hamiltonian, and the power counting of the effective theoryprovides us with theoretical uncertainty estimates. We successfully test theeffective theory in homonuclear molecules that exhibit a large separation ofscales. For ground-state band transitions of rotational nuclei, the effectivetheory describes data well within theoretical uncertainties at leading order.In order to probe the theory at subleading order, data with higher precisionwould be valuable. For transitional nuclei, next-to-leading order calculationsand the high-precision data are consistent within the theoretical uncertaintyestimates. We also study the faint inter-band transitions within the effectivetheory and focus on the $E2$ transitions from the $0^+_2$ band (the "$\beta$band") to the ground-state band. Here, the predictions from the effectivetheory are consistent with data for several nuclei, thereby proposing asolution to a long-standing challenge.

Effective theory for the non-rigid rotor in an electromagnetic field: Toward accurate and precise calculations of E2 transitions in deformed nuclei