Correlation length of the angular mode for an approximate $U(1)$ symmetry during inflation

It is known that a light scalar field obtains fluctuations in the de Sitter inflationary background. Such fluctuations could provide an initial condition for baryogenesis through the Affleck-Dine mechanism, where an approximate $U(1)_B$ symmetry is usually assumed. However, an interpretation of the baryon number generation in this way is strongly related to the correlation length of the angular mode. In this work, we calculate the correlation length of the angular mode for a model exhibiting an approximate $U(1)$ symmetry. We find that for a massive nearly non-interacting field, the correlation length of the angular mode is determined by the mass parameter of the model and it is similar to $H^{-1} \exp(H^2/m^2)$. Applying this result to baryogenesis via the Affleck-Dine mechanism with a stochastic origin, we find that only for $m \ll \mathcal O(0.1) H$(assuming $N_*=60$) can the correlation length of the baryon number density be much larger than our current horizon size, such that we live in the baryon-rich region. If this is not true, in the early time our universe would consist of numerous patches of baryon-rich and anti-baryon-rich regions with the average baryon number being nearly zero.