Analytic determination of the regularizing black-ring parameter

Determine an analytical expression for the value \(\lambda_c\) that satisfies the conical-regularity condition \(\mu=1\) for the swirling black ring, as a function of \(\nu\), \(R\), and \(j\).

Background

The swirling black ring is regular only if the conical-singularity condition $\mu=1$ is imposed, which fixes the parameter $\lambda$ to a value denoted by $\lambda_c$. The authors recover the known asymptotically flat value when $j=0$ and report numerical evidence that a regularizing value exists for at least some parameter ranges.

An explicit analytical formula for $\lambda_c$ would characterize the regular parameter space of the swirling black-ring solution and clarify the ranges in which the conical defects can be removed.