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A new family of Barnes beta distributions on $(0, \infty)$ is introduced andits infinite divisibility, moment determinacy, scaling, and factorizationproperties are established. The Morris integral probability distribution isconstructed from Barnes beta distributions of types $(1,0)$ and $(2,2),$ andits moment determinacy and involution invariance properties are established.For application, the maximum distributions of the 2D gaussian free field on theunit interval and circle with a non-random logarithmic potential areconjecturally related to the critical Selberg and Morris integral probabilitydistributions, respectively, and expressed in terms of sums of Barnes betadistributions of types $(1,0)$ and $(2,2).$

On Barnes Beta Distributions and Applications to the Maximum Distribution of the 2D Gaussian Free Field