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The spin response functions measured in multi-component fermion gases bymeans of rf transitions between hyperfine states are strongly constrained bythe symmetry of the interatomic interactions. Such constraints are reflected inthe spin f-sum rule that the response functions must obey. In particular, onlyif the effective interactions are not fully invariant in SU(2) spin space, arethe response functions sensitive to mean field and pairing effects. Wedemonstrate, via a self-consistent calculation of the spin-spin correlationfunction within the framework of Hartree-Fock-BCS theory, how one can derive acorrelation function explicitly obeying the f-sum rule. By contrast, simpleone-loop approximations to the spin response functions do not satisfy the sumrule. As we show, the emergence of a second peak at higher frequency in the rfspectrum, as observed in a recent experiment in trapped $^6\text{Li}$, can beunderstood as the contribution from the paired fermions, with a shift of thepeak from the normal particle response proportional to the square of the BCSpairing gap.Comment: 7 pages, 1 figure, content adde

Spin-correlation functions in ultracold paired atomic-fermion systems: Sum rules, self-consistent approximations, and mean fields