Probing the three-body force in hadronic systems with specific charge parity

Three-body forces, a type of non-perturbative strong interaction, are widely studied in nuclear physics. However, whether their inclusion is necessary in nuclear systems remains a topic of intense debate. In this letter, we propose that the existence of three-body forces in certain three-body hadronic systems with definite $C$-parity is certain. Such systems consist of two components whose interactions are mediated by three-body forces--a mechanism not easily realized in conventional three-nucleon systems. We investigate two specific three-body hadronic systems, $\bar{D}_sDK$ and $\bar{D}^*D\eta$, using contact-range potentials. The two-body hadron-hadron interactions are constrained by reproducing their scattering lengths, while the three-body couplings are constrained by charge symmetry. Our results indicate that three-body forces play a minor role in binding the $I(J^{PC})=0(0^{--})$ $\bar{D}_sDK$ system, but a crucial one in binding the $I(J^{PC})=0(1^{-+})$ $\bar{D}^*D\eta$ system. In fact, three-body forces determine whether $\bar{D} ^*D\eta$ forms a bound state, making this system a promising candidate for exploring three-body forces in hadronic physics.