An Instability in the Matrix Solution of DGLAP Equations

Following, the matrix solution outlined by Ratcliffe, and using data points (g1p,n,d or) F2p,n,d(xi,Qij2) as the only parameters -thus doing away with the need for a set of free parameters used in a phenomenological fit to (polarized) parton distributions usually adopted (GRV, MRS, CTEQ, ...)- we arrive at a system of linear equations. Taking the unpolarized case, where there is plenty of data, and a possible combination of unknowns q8, Σ, and g, where q3(xi,Qi12) 3F2(p-n)(xi,Qi12) can be known, there remains three linear equations with three unknowns at every xi, each decomposing one of the data points F2p(xi,Qij2), j = 1,2,3. The unknowns would be at (xi,Qi12), since the matrix solution of the nonsinglet and singlet DGLAP evolution equations can be used to determine the unknowns at (xi,Qij2), j = 2,3 in terms of those at (xi,Qi12). The determinant of coefficients of these equations is too small to permit a stable solution within the range of the errors of F2p values. The significance of the problem and possible solutions for it are discussed