There is mounting evidence that as many as 1000 neutron stars (NSs) may be present in some of the richest globular clusters in the Galaxy, which perhaps amounts to more than 10%-20% of the NSs ever formed in each cluster. Such a large NS retention fraction is seemingly at odds with recent estimates of the characteristic kick speeds of single radio pulsars in the Galaxy, ranging from roughly 5 to 10 times the central escape speeds of the most massive globular clusters. This retention problem is a long-standing mystery. It has been suggested that the retention problem may be solved if one assumes that a large fraction of NSs in clusters were formed in binary systems. We present a thorough investigation of this possibility that involves a population study of the formation and evolution of massive binary systems. We utilize a Monte Carlo approach to generate and evolve an ensemble of massive primordial binaries. Binary component masses and orbital parameters are chosen from appropriate distribution functions. Analytic prescriptions are used to evolve the masses and the orbital separation during any important episodes of mass transfer, whether the mass transfer is stable or dynamically unstable. The eventual collapse and supernova explosion of the core of the initially more massive star is accompanied by sudden mass loss and an impulsive kick to the newly formed NS. A very straightforward and general mathematical formalism is used to compute the new orbital parameters immediately after the supernova or the speeds of the individual components if the binary is ionized. If we apply the large mean NS kick speeds inferred from pulsar observations, we find that most binaries are unbound following the supernova, and all but a very small fraction of the liberated NSs are ejected from the cluster. As expected, the majority of retained NSs have massive companions, which are mostly the products of stable mass transfer. Systems that undergo dynamically unstable mass transfer shrink dramatically and acquire large relative orbital speeds ( typically greater than or similar to 200 km s(-1)). The combined effects of sudden mass loss and the NS kick in the subsequent supernova explosion lead to the escape of many of these binaries. Our standard model involving the formation of NSs in binary systems predicts that 5% of the NSs initially formed in a massive cluster can be retained. Over a wide range of model parameters, the retention fraction varies from similar to1% to 8%. When a number of other effects are taken into account, e.g., a reasonable binary fraction among massive stars, the retention fraction may become several times smaller. Therefore, we suggest that perhaps the conventional thinking regarding NS kicks must be modified or that a new paradigm must be adopted for the evolution of some of the most massive globular clusters in the Galaxy.
