Scaling of temporal correlations in an attractive Tomonaga-Luttinger spin liquid

We report temperature-dependent neutron scattering measurements of the local dynamic structure factor in the quantum spin ladder (C${}_7$H${}_{10}$N)${}_2$CuBr${}_4$ in a magnetic field $H=9$~T, in its gapless quantum-critical phase. We show that the measured quantity has a scaling form consistent with expectations for a Tomonaga-Luttinger liquid with attraction. The measured Luttinger parameter $K\approx 1.25$ and scaling function are in excellent agreement with density matrix renormalization group numerical calculations for the underlying spin Hamiltonian.