Better criterion for morphing basis optimization

[johannbrehmer]

Currently we optimize the morphing basis (benchmark points) by minimizing the expected squared morphing weights L[w] = E_theta[sum_i w_i(theta)^2]. The expectation runs over a flat prior over the parameter ranges specified in MadMiner.add_parameter(). Generally, our current algorithm will push the benchmark points far away from each other, which does indeed typically lead to small morphing weights.

However, the squared weights aren't really the same as the true morphing error. The true morphing error will also depend on how much the differential cross section changes over the parameter space. When all weights are numerically small, but the different cross sections for each benchmark are wildly different, the morphing error will be large even though L[w] is small. Of course, the cross sections will be more similar for closer parameter points, so this effect is opposite to the effect from the morphing weights.

The problem is of course that the differential cross section is process-dependent. At the stage where we have to pick the morphing basis, we don't have any numerical results yet.

So how can we improve this? One option would be to generate a small event sample with some morphing basis first and then to use this to calculate actual morphing errors as a function of a morphing basis. But that's a bit more complicated. Any other ideas?