Antisymmetric part of the dynamic structure function of liquidHe4

At large momentum transfers it is convenient to express the dynamic structure function $S(k, \omega )$ as the sum of a symmetric part about $\omega =k^2$ and an antisymmetric part. The latter is zero in the impulse approximation, and its leading contribution is given by $\frac{S_A\mathrm{}\left(y\right)\mathrm{}}{{\mathrm{}\left(2k\right)\mathrm{}}^2}$, where $y=\frac{(\omega -k^2)}{2k}$ is the usual scaling variable. We calculate the integrals of $S_A\mathrm{}\left(y\right)\mathrm{}$, weighted with $y$, $y^3$, and $y^5$ in liquid ${}_{}{}^4{}_{}{}^{}\mathrm{He}$ using sum rules as suggested by Sears. Polynomial expansions are used to construct models of $S_A\mathrm{}\left(y\right)\mathrm{}$ which appear to be in qualitative agreement with the observed antisymmetric part at large values of $k$.