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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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I am not familiar with the area but in ref. 4 there are plenty of derivations given (e.g. eqns. (1), (3), (84)). I guess integrating one or some of the eqns. could give you your equation. But I really just browsed through the articles and I am not even sure if their notation matches yours or the ones in the newer articles. Just to have an idea how this may come from.

EDIT: I found another paper in the reference that might clear things a little - Link to the paper. In II. D. you will find the following statement:

Finally we have to specify the initial condition (i.c.) for the evolution or, equivalently, the precise shape of the proton unintegrated gluon distribution (UGD), ...

So what I guess is that it is not that easy to alter the initial conditions to the UGD. I think a more or less complicated parametrization is needed to do so. The paper referenced in that connection is this one: Link to the paper.

I hope this helps a little.

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