Statistics
Beta is the correlation coefficient between two variables measured in standardized units; beta = 0 indicates zero correlation because standardized slopes and correlations are the same numeric value in all cases.
Beta is the sample slope estimate computed from observed data; beta = 0 means the observed regression line is flat in the sample and thus the data cannot have any variation.
Beta is the change in the population mean of the response per one-unit change in the explanatory; beta = 0 means that changes in the explanatory do not produce any expected change in the response, so there is no linear relationship.
Beta is the variance of the residuals from the regression line; beta = 0 means there is no variance and therefore no linear correlation.