# What is the first appearance of the MV (McLerran-Venugopalan) initial condition?

First a quick introduction for the unfamiliar: in saturation physics (my research field), a lot of theoretical work centers on the BK (Balitsky-Kovchegov) equation, which is a differential equation which governs the structure of the proton. It basically takes the form

$$\frac{\partial}{\partial Y}N = K\otimes N - N^2$$

$N$ is a function related to the proton's structure, and is what we solve the equation for. It's a function of $\mathbf{r}$, the position in the plane perpendicular to the beam line, and of $Y = -\ln x$, where $x$ is the momentum fraction of the quark or gluon involved in the collision. $K\otimes$ is some integral operator.

Solving an equation of this form starts with an initial condition at some initial $Y = Y_0$. I've seen a number of recent papers (1,2,3,etc.) that use an initial condition of this form:

$$N(\mathbf{r}, Y_0) = 1 - \exp\biggl[-\frac{(r^2 Q_{s0}^2)^\gamma}{4}\ln\biggl(e + \frac{1}{r\Lambda}\biggr)\biggr]$$

Usually the equation is accompanied by the citation of a trio of papers (4,5,6) by Larry McLerran and Raju Venugopalan from 1994, and accordingly the expression is called the MV initial condition. The thing is, I checked those papers from 1994 and I can't seem to find this expression anywhere in them, nor can I find anything that it can obviously be derived from. So I'm wondering, is there a later paper that actually derives or postulates the MV initial condition itself? Perhaps by starting from the results of the McLerran and Venugopalan papers?

Or is it really in one or more of those three 1994 papers, and I'm just missing it?

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