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In the past few years, several hints of lepton flavour universality (LFU)violation have emerged in the $b \to c \tau \bar\nu$ and $b \to s \ell^+\ell^-$ data. Quite recently, the Belle Collaboration has reported the firstmeasurement of the $D^*$ longitudinal polarization fraction in the $B \to D^*\tau \bar\nu$ decay. Motivated by this intriguing result, together with therecent measurements of $R_{J/\psi}$ and $\tau$ polarization, we study $b \to c\tau \bar\nu$ decays in the Supersymmetry (SUSY) with $R$-parity violation(RPV). We consider $B \to D^{(*)} \tau \bar\nu$, $B_c \to \eta_c \tau \bar\nu$,$B_c \to J/\psi \tau \bar\nu$ and $\Lambda_b \to \Lambda_c \tau \bar\nu$ modesand focus on the branching ratios, the LFU ratios, the forward-backwardasymmetries, polarizations of daughter hadrons and $\tau$ lepton. It is foundthat the RPV SUSY can explain the $R_{D^{(*)}}$ anomalies at $2\sigma$ level,after taking into account various flavour constraints. In the allowed parameterspace, the differential branching fractions and LFU ratios are largely enhancedby the SUSY effects, especially in the large dilepton invariant mass region. Inaddition, a lower bound $\mathcal B(B^+ \to K^+ \nu \bar\nu) > 7.37 \times10^{-6}$ is obtained. These observables could provide testable signatures atthe High-Luminosity LHC and SuperKEKB, and correlate with direct searches forSUSY.

$b \to c\tau\bar\nu$ decays in supersymmetry with $R$-parity violation