My profound apologies if this question is ill posed. I am not entirely sure myself. Here is the fact. Often in nuclear physics we encounter partial differential distributions. Let's make it simple.
I have a particle with momentum $p_{\text{inc}}$ along the $z$ axis that impinges on "something" and produces two particles with momenta and angle $(p_1,\theta_1),(p_2,\theta_2)$. Let's suppose that we have a PDF of this kind describing the joint distribution:
\begin{equation} F(p_{\text{inc}},p_1, \theta_1, p_2, \theta_2) = \frac{\partial^4 \sigma(p_{\text{inc}},p_1, \theta_1, p_2, \theta_2)}{\partial p_1 \partial \theta_1 \partial p_2 \partial \theta_2} \end{equation}
In this case we suppose a $\phi$ symmetry. Unfortunately in the data I only have:
\begin{equation} F_{i}(p_{\text{inc}},p_i, \theta_i) = \int_{0}^{\pi}d\theta_{3-i}\int_{0}^{\infty}{dp_{3-i}\frac{\partial^4 \sigma(p_{\text{inc}},p_1, \theta_1, p_2, \theta_2)}{\partial p_1 \partial \theta_1 \partial p_2 \partial \theta_2}} \end{equation}
If I am writing a MonteCarlo, I can of course sample from the two partial PDFs, but the values of $p_i$ and $\theta_i$ will be uncorrelated and they will not obey to the energy-momentum conservation:
\begin{equation} \overrightarrow{p_{\text{inc}}}-\sum_{i=1}^2\overrightarrow{p_{i}}=0 \end{equation}
Obviously I cannot reconstruct $F(p_{\text{inc}},p_1, \theta_1, p_2, \theta_2)$, the information lost is lost. However I would like to know whether I can somehow find a (family of) PDFs from which I can sample $(p_1,\theta_1),(p_2,\theta_2)$ that would give me the right partial PDFs and that would respect the energy-momentum conservation.
Were this possible, it would be then reasonable to start considering what other constraints to impose on this family of PDFs (ideally parametric) to obtain a "gentle guess" of the real thing. This may seem very approximate, but at the moment we are left with sampling from the two PDFs and "hope" that "on average" energy and momentum are conserved.
I.e. the bar is very low. Thanks a lot for your attention and sorry again if this is confusing or plain stupid.
All help welcome! Best,
Federico Carminati