  Inelastic neutron scattering is employed to study transverse spin
correlations of a Heisenberg $S=1/2$ chain compound in a magnetic field of 7.5
T. The target compound is the antiferromagnetic Heisenberg $S=1/2$ chain
material 2(1,4-Dioxane)$\cdot$2(H$_2$O)$\cdot$CuCl$_2$, or CuDCl for short. The
validity and the limitations of the scaling relation for the transverse dynamic
structure factor are tested, discussed and compared to the Tomonaga-Luttinger
spin liquid theory and to Bethe-ansatz results for the Heisenberg model.
