  We investigate the theoretical framework of the $\cos 2\phi$ azimuthal
asymmetry contributed by the coupling of two Boer-Mulders functions in the
dilepton production unpolarized $\pi p$ Drell-Yan process by applying the
transverse momentum dependent factorization at leading order. We adopt the
model calculation results of the unpolarized distribution function $f_1$ and
Boer-Mulders function $h_1^\perp$ of pion meson from the light-cone wave
functions. We take into account the transverse momentum evolution effects for
both the distribution functions of pion and proton by adopting the existed
extraction of the nonperturbative Sudakov form factor for the pion and proton
distribution functions. An approximate kernel is included to deal with the
energy dependence of the Boer-Mulders function related twist-3 correlation
function $T_{q,F}^{(\sigma)}(x,x)$ needed in the calculation. We numerically
estimate the Boer-Mulders asymmetry $\nu_{BM}$ as the functions of $x_p$,
$x_\pi$, $x_F$ and $q_T$ considering the kinematics at COMPASS Collaboration.
