  We present the high-precision magnetization data of the valence fluctuating
heavy fermion superconductor $\beta$-YbAlB$_4$ in a wide temperature range from
0.02 K to 320 K spanning four orders of magnitude. We made detailed analyses of
the $T/B$ scaling of the magnetization, and firmly confirmed the unconventional
zero-field quantum criticality (QC) without tuning. We examined other possible
scaling relationship such as $T/(B-B_c)^{\delta}$ scaling, and confirmed that
$\delta = 1$ provides the best quality of the fit with an upper bound on the
critical magnetic field $\vert B_c \vert <0.2$~mT. We further discuss the heavy
Fermi-liquid component of the magnetization after subtracting the QC component
estimated based on the $T/B$ scaling. The temperature dependence of the heavy
Fermi-liquid component is found very similar to the magnetization of the
polymorph $\alpha$-YbAlB$_4$. In addition, the heavy Fermi-liquid component is
suppressed in the magnetic field above $\sim$ 5 T as in $\alpha$-YbAlB$_4$.
This was also confirmed by the magnetization measurements up to $\sim 50$ T for
both $\alpha$- and $\beta$-YbAlB$_4$. Interestingly, the detailed analyses
revealed that the only a part of $f$ electrons participates in the zero-field
QC and the heavy fermion behavior. We also present a temperature - magnetic
field phase diagram of \ybal to illustrate how the characteristic temperature
and field scales evolves near the QC.
