  At present, there are several measurements of $B$ decays that exhibit
discrepancies with the predictions of the Standard Model, and suggest the
presence of new physics in $b\to s$ and $b \to c(u)$ quark level transitions.
Motivated by the prospects of the ongoing high-luminosity $B$ factories, we
study the exclusive $B \to a_1 (1260) \ell^- \bar{\nu}_\ell$ process within the
Standard Model Effective Field Theory (SMEFT) formalism, to understand the
sensitivity of new physics. The new physics parameters are constrained by using
the experimental branching fractions of the (semi)leptonic $B \to \ell
\bar{\nu}$ and $B \to (\pi, \rho, \omega) \ell \bar{\nu}$ processes (where
$\ell = e, \mu, \tau$) which undergo $b \to u \ell \bar{\nu}$ quark level
transitions. We then perform a comprehensive angular analysis of the exclusive
$B \to a_1 (1260) \ell^- \bar{\nu}_\ell$ process in the Standard Model and in
the presence of various new physics operators. We also provide the predictions
and comment on various observables, such as branching ratio, forward-backward
asymmetry, and the test of lepton flavor non universality of the $B \to a_1
(1260) \ell^- \bar{\nu}_\ell$ channel.
