  Accretion disks surrounding stellar mass black holes (BHs) have been
suggested as potential locations for the nucleosynthesis of light elements,
which are our primary observational discriminant of multiple stellar
populations within globular clusters. The population of enriched stars in
globular clusters are enhanced in N14, Na23, and sometimes in Al27 and/or in
K39. In this study, our aim is to investigate the feasibility of initiating
nucleosynthesis for these four elements in BH accretion disks, considering
various internal parameters such as the temperature of the gas and timescale of
the accretion. To achieve this, we employed a 132-species reaction network. We
used the slim disk model, suitable for the Super-Eddington mass accretion rate
and for geometrically and optically thick disks. We explored the conditions
related to the mass, mass accretion rate, viscosity, and radius of the
BH-accretion disk system that would allow for the creation of N14, Na23, Al27,
and K39 before the gas is accreted onto the central object. Our findings reveal
that there is no region in the parameter space where the formation of Na23 can
occur and only a very limited region where the formation of N14, Al27, and K39
is plausible. Specifically, this occurs for BHs with masses lower than 10 solar
masses, with a preference toward even lower mass values and extremely low
viscosity parameters ($\alpha <10^{-3}$). Such values are highly unlikely based
on current observations of stellar mass BHs. However, such low mass BHs could
actually exist in the early universe, as so-called primordial BHs. In
conclusion, our study suggests that the nucleosynthesis within BH accretion
disks of four elements of interest for the multiple stellar populations is
improbable, but not impossible, using the slim disk model.
