How to transform a Hessian set into the corresponding Monte Carlo PDF

A Hessian PDF set can be transformed into a Monte Carlo replica set using the method described in Eq. (4.3) of PDF4LHC21 using a runcard like the one below. In this example Neig are the number of basis eigenvectors of the Hessian PDF. mc_pdf_name is the name of the new MC replica set that will be created. watt_thorne_rnd_seed is the random seed used to generate the MC replicas as shown in the equation below.

\[f^{MC, (k)}(x,Q) = f^{H}_{0}(x,Q) + \frac{1}{2} \sum_{j=1}^{Neig} \left(f^{H}_{j,+}(x,Q) - f^{H}_{j,-}(x,Q)\right) R_j^{(k)}\]

where \(f^{MC, (k)}(x,Q)\) is the \(k\)-th Monte Carlo replica, \(f^{H}_{0}(x,Q)\) the central Hessian member, \(f^{H}_{j,\pm}(x,Q)\) the positive and negative eigenvector directions corresponding to the \(j\)-th Hessian eigenvector, and \(R_j^{(k)}\) are random standard normal numbers.

The new MC replica set will be saved in the LHAPDF folder.

pdf: MSTW2008nlo68cl

mc_pdf_name: MSTW2008nlo68cl_mc

num_members: 100

watt_thorne_rnd_seed: 1

actions_:
- write_hessian_to_mc_watt_thorne