  We propose that the transverse momentum ($p_T$) differential splitting of
directed flow ($\Delta v_1$) between proton and anti-proton can serve as a
sensitive observable to extract the baryon diffusion coefficient ($\kappa_B$)
of the hot and dense strongly interacting matter produced in relativistic heavy
ion collisions. We use relativistic dissipative hydrodynamics framework with
Glauber model based initial condition for the energy as well as baryon
deposition that is calibrated to capture the rapidity dependence of charged
particle multiplicity, net proton yield as well as the elusive $v_1$ splitting
between proton and anti-proton. We employ the commonly used kinetic theory
motivated ansatz: $\kappa_B= C_B \frac{n_B}{T} \left( \frac{1}{3}
\text{coth}\left(\frac{\mu_B}{T} \right) -
  \frac{n_BT}{\epsilon+P} \right)$ where $n_B$, $\epsilon$, $P$, $T$ and
$\mu_B$ are baryon number density, energy density, pressure, temperature and
baryon chemical potential respectively while $C_B$ is an arbitrary constant
which is largely unknown for the Quantum Chromodynamics (QCD) medium. We find
that the variation of $\Delta v_1$ with $p_T$ is strongly influenced by the
choice of $C_B$. Further, we find that the recent STAR measurement of the
centrality dependence of the rapidity slope of $\Delta v_1$ prefers
$0.5<C_B<1.5$.
