The gauge field is really a vector field, a transformed vector field from the integral of both relative vector field and gradient of relative scalar field.

However, field measurability must always be scalar. This is the results of almost all experimental investigations. On the other hand if directional property is desired then unavoidably a frame of reference must always be defined since nature relatively has no preferred direction of motion. However, nature is still absolutely covariance, the forms of its physical description in the language of mathematics is always the same in any coordinate system or frame of reference.