  Recent experimental results of ${\cal R}(D^{(*)})$ deviate from the standard
model (SM) by $3.1\sigma$, suggesting a new physics (NP) that affects the $b\to
c \tau \bar\nu_\tau$ transition. Motivated by this, we investigate the possible
NP effects in the $\Lambda_b\to\Lambda_c \tau\bar\nu_\tau$ decay. For this
purpose, assuming the neutrinos are left-handed, we calculate in detail the
helicity amplitudes of $\Lambda_b\to\Lambda_c \ell\bar\nu_\ell$ decays with all
possible four-fermion operators. Within the latest results of
$\Lambda_b\to\Lambda_c$ form factors from lattice QCD calculations, we study
these decays in a model-independent manner. The differential and total
branching fractions and other observables are calculated. In SM, we obtain the
ratio ${\cal R}(\Lambda_c)=0.33\pm0.01$. Supposing that NP only affects the
third generation fermions, we present the correlations among ${\cal R}(D)$,
${\cal R}(D^*)$ and ${\cal R}(\Lambda_c)$. We perform a minimum $\chi^2$ fit of
the wilson coefficient of each operator to the latest experimental data of
different observables. It is found that the left-handed scalar operator ${\cal
O}_{SL}$ affects the branching fraction remarkably, and the ratio ${\cal
R}(\Lambda_c)$ can be enhanced by $30\%$. For other operators, the ratio
amounts to $0.38\pm0.02$, which is larger than prediction of SM by $20\%$.
Using the fitted values of the wilson coefficients of the single NP operators,
we also give a prognosis for the physical observables of $\Lambda_b\to\Lambda_c
\tau\bar\nu_\tau$. Furthermore, we also study the effects of three typical NP
models on $\Lambda_b\to\Lambda_c \tau\bar\nu_\tau$. We hope our results can be
tested in the current LHCb experiment and the future high energy experiments.
