Asymmetric Gluon Distributions and Hard Diffractive Electroproduction

The ``asymmetric'' matrix element $\langle p-r | G \ldots G |p \rangle$ that appears in the pQCD description of hard diffractive electroproduction does not coincide with that defining the gluon distribution function $f_g(x)$. I outline a pQCD formalism based on a concept of the double distribution $F_g(x,y)$, which specifies the fractions $xp$, $yr$, $(1-y)r$ of the initial proton momentum $p$ and the momentum transfer $r$, $resp.,$ carried by the gluons. For $t \equiv r^2 =0$, $r$ is proportional to $p$: $r = \zeta p$, and it is convenient to parameterize the matrix element $\langle p-r | G \ldots G |p \rangle$ by an asymmetric distribution function ${\cal F}_{\zeta}^g (X)$ depending on the total fractions $X \equiv x+y \zeta$ and $X-\zeta = x- (1-y) \zeta$ of the initial hadron momentum $p$ carried by the gluons.I formulate evolution equations for ${\cal F}_{\zeta}^g (X)$, study some of their general properties and discuss the relationship between ${\cal F}_{\zeta}^g (X)$, $F_g(x,y)$ and $f_g(x)$.