  The radiative branching ratio of the Hoyle state is crucial to estimate the
triple-$\alpha$ reaction rate in stellar environments at medium temperatures.
Knowledge of the $\gamma$-decay channel is critical as this is the dominant
radiative decay channel for the Hoyle state. A recent study by Kib\'edi et al.
[Phys. Rev. Lett. 125, 182701 (2020)] has challenged our understanding of this
astrophysically significant branching ratio and its constraints. The objective
of this work was to perform a new measurement of the $\gamma$-decay branching
ratio of the Hoyle state to deduce the radiative branching ratio of the Hoyle
state. An additional objective was to independently verify aspects of the
aforementioned measurement conducted by Kib\'edi et al. For the main experiment
of this work, the Hoyle state was populated by the $^{12}\textrm{C}(p,p')$
reaction at 10.8 MeV at the Oslo Cyclotron Laboratory. The $\gamma$-decay
branching ratio was deduced through triple-coincidence events, each consisting
of a proton ejectile corresponding to the Hoyle state, and the subsequent
$\gamma$-ray cascade. In the main experiment of this work, a $\gamma$-decay
branching ratio of the Hoyle state of $\Gamma_{\gamma}/\Gamma=4.0(4)\times
10^{-4}$ was determined, yielding a corresponding radiative branching ratio of
$\Gamma_{\textrm{rad}}/\Gamma=4.1(4) \times 10^{-4}$, which is in agreement
with several recent studies, as well as the previously adopted ENSDF average of
$\Gamma_{\textrm{rad}}/\Gamma=4.16(11)\times 10^{-4}$. Aspects of the analysis
performed by Kib\'edi et al. were verified in this work and the source of
discrepancy between the results of this work and that of Kib\'edi et al. could
not be determined. Further independent and innovative studies for the radiative
width of the Hoyle state will substantiate whether the discrepant result by
Kib\'edi et al. should be excluded from future evaluations.
