Student distribution for correlation of Gaussian samples

Question

Given indpendent random variables $X_1 , \ldots , X_n , Y_1 , \ldots , Y_n$ where $X_i$ have Gaussian probability distribution with mean $\mu_1$ and standard deviation $\sigma_1$, while $Y_i$ have Gaussian probability distribution with mean $\mu_2$ and standard deviation $\sigma_2$, let us consider the random variable $$ C := \frac{ \tfrac{1}{n} \sum_{i=1}^n \left( (X_i - \tfrac{1}{n} \sum_{j=1}^n X_j) (Y_i - \tfrac{1}{n} \sum_{j=1}^n Y_j) \right) }{ \sqrt{ \tfrac{1}{n}\sum_{i=1}^n ( X_i - \tfrac{1}{n}\sum_{j=1}^n X_j )^2 } \sqrt{ \tfrac{1}{n}\sum_{i=1}^n ( Y_i - \tfrac{1}{n}\sum_{j=1}^n Y_j )^2 } }.$$ Probably one should normalize $C$ by some coefficient depending on $n$ to make things look nicer. What is the probability distribution of $C$ (normalized appropriately)? Is there a reference? Probably it should be related to Student's $\mathbf{t}$-distribution.