Analytic solution

In case only linear EFT corrections are needed or available, SMEFiT allows you to obtain the analytic solution of the linear problem.

The central value of the Wilson coefficient is given by solving the equation \(\partial \chi^2 / \partial c = 0\):

\[c_i = \left ( \kappa_{i,k} \text{cov}_{k,l}^{-1} \kappa_{j,l} \right )^{-1} \kappa_{j,m} \text{cov}_{m,n}^{-1} \left ( \sigma_{n}^{\text{(exp)}} - \sigma_{n}^{\text{(sm)}} \right ) \quad i,j=\{1,\dots N_{op}\}, \quad m,n,k,l=\{1,\dots N_{dat}\}\]

and its covariance matrix is given by the Fisher information:

\[X_{ij} = \kappa_{i,k} \text{Cov}_{k,l}^{-1} \kappa_{j,l} \quad i,j=\{1,\dots N_{op}\}\]

Starting from the central values and the covariance one can then draw the required number of samples from a muli Gaussian distribution \(\mathcal{N}(c, X)\).

This analytic method is really efficient, however it does not work in the presence of flat directions, which can result in a \(X_{ij}\) matrix that is not semipositive definite.

Also quadratic relations between Wilson coefficient, as well as quadratic EFT corrections are not supported.