Degenerate soft modes and selective condensation in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>BaAl</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:msub> <mml:mi mathvariant="normal">O</mml:mi> <mml:mn>4</mml:mn> </mml:msub> </mml:mrow> </mml:math> via inelastic x-ray scattering

BaAl$_2$O$_4$ is a ferroelectric material that exhibits structural quantum criticality through chemical composition tuning. Although theoretical calculations and several diffraction experiments have suggested the involvement of a soft mode in its ferroelectric structural phase transition, direct experimental verification is still lacking. In this study, we successfully observed two soft modes of BaAl$_2$O$_4$ using x-ray inelastic scattering, providing direct experimental evidence for their role in the structural phase transition. Furthermore, we reveal that the soft modes at the M and K points are nearly degenerate in energy, indicating a delicate balance in which either mode could potentially freeze. The K-point mode simultaneously softens toward the transition temperature ($T_{\rm C}$) in a manner nearly identical to the M-point mode. However, the phase transition condenses only at the M point, with the M-point mode stabilizing as an acoustic mode in the low-temperature structure and the K-point mode hardening as temperature decreases.