Correlations between inclusive DIS variables

Page maintainer: S. Gliske

This page is considered done. It been reviewed by Henk/Caro. There may be missing elements, but they are all flagged and the text has no errors.

Intro

Of the Lorentz invariant inclusive DIS variables $\left.x,y,Q^{2},W^{2}\right.$, (see kinematic variable definitions) one can always choose two as independent quantities. The other two can then be expressed as a function of the chosen two. In this sense the chosen two are independent. However, much discussion has taken place regarding whether, and in what manner, the chosen two are yet correlated. This page is intended to summarize the discussions.

Dependent vs. Independent Variables

For concreteness, let the chosen independent variables be $x$ and $y$. We can then write

      $\left.Q^{2}=2xyME\right.$
      $\left.W^{2}=M^{2}+2M\nu -Q^{2}\right.$

where $\left.M\right.$ is the target mass and $\left.E\right.$ is the beam energy. To confuse the matter, sometimes one writes the differential cross section with respect to the chosen variables as a function of both the independent and dependent variables

      ${\frac {d^{2}\sigma }{dxdy}}=f(x,y,Q^{2},W^{2}).$

In this case $\left.Q^{2}\right.$, $\left.W^{2}\right.$, are understood as functions of $\left.x\right.$ and $\left.y\right.$.

Emergent Correlations due to Cuts and Binning

Consider a uniform distribution in $\left.x\right.$ and $\left.y\right.$, in some rectangular region $\left.x_{min}<x<x_{max}\right.$, $\left.y_{min}<y<y_{max}\right.$. If one were to consider the $\left.x\right.$ projection of this distribution, one would also see a uniform distribution.

Now consider placing additional cuts on $\left.Q^{2}\right.$ and $\left.W^{2}\right.$. In this case, the valid domain of the PDF is no longer rectangular, but oddly shapped. For concreteness, see this plot from bootcamp. Although this is actually the $\left.x\right.$ vs. $\left.Q^{2}\right.$ plot, the concept is the same. Projecting a uniform distribution in $\left.x\right.$ and $\left.y\right.$, but within the $\left.Q^{2}\right.$ and $\left.W^{2}\right.$ cuts, would result in a distribution that looks non-uniform. This is because the integration range is oddly shaped.

Furthermore, for any distribution in $\left.x\right.$ and $\left.y\right.$, a $\left.Q^{2}>Q_{min}^{2}\right.$ cut will remove data from the low $\left.x\right.$, low $\left.y\right.$ region, making it appear in projections that $\left.x\right.$ and $\left.y\right.$ are correlated.

Correlations due to Acceptance

It is possible that acceptance could cause perceived correlations. This occurs in exactly the same manner above. Just as placing cuts on the dependent variables changes the shape of the valid domain of the independent variables, the shape of the spectrometer also influences the valid domain of the independent variables. However, for inclusive DIS variables, the main cause of perceived correlations is actually the cuts, not the spectrometer shape.

Correlations due to the PDFs

It is also possible that the cross section has terms causing correlations. For instance, the SIDIS cross-section for pseudo-scaler meson production has a prefactor of $\left.1/(xy)\right.$, with additional $\left.x\right.$, $\left.y\right.$, and $\left.Q^{2}\right.$ dependencies elsewhere. This prefactor likewise causes data to be denser when $\left.x\right.$ and $\left.y\right.$ are both small, which could be seen as a correlation.

Summary

Although one can always choose two of the four Lorentz inclusive DIS variables as independent, cuts on the dependent variables and the spectrometer shape will change the shape of the valid domain of the independent variables. This non-rectangular shape then appears as correlations in projections and when binning. In addition, the actual cross section may define further correlations. However, the primary effect is due to cutting on the dependent variables.