Resummation and Simulation of Soft Gluon Effects beyond Leading Color

Towards colour flow evolution at two loops

“…(2.1) can only proceed by using Monte Carlo methods to sample over the different colour structures involved, possibly paired with approximation methods to keep track of certain classes of 1/N suppressed contributions to the cross section [5,14]. From a computational point of view, the colour flow basis has proven to be useful in such an approach, and a practical code has been based on it [15]. In order to be able to solve the evolution equations in this approach at the next order, we choose to express the anomalous dimension and emission matrices not only in the basis independent notation, but also in the colour flow basis.…”

“…While, in a perturbative expansion, one would treat the diagonal, 1/N 2 suppressed bit as a correction this turns out not to be a viable approach in the presence of collinear contributions: in this case, dropping the 1/N 2 contribution amounts to effectively replacing C F by C A /2 in the quark splitting function, which would thus not properly take into account logarithmic contributions of soft-and hard-collinear origin. For this reason we stress that an appropriate expansion around the large-N limit would actually need to be seen as an expansion around the colour diagonal part (also referred to as d approximations in [15]), something which will be of great importance when discussing the colour structures appearing the two-loop case. Also note that the 1/N 2 suppression in the one-loop anomalous dimension might possibly be overcome since ρ contains a sum over all pairs of quarks and antiquarks, not only colour connected dipoles.…”

“…(A.3)). The results of the colour structures can be used for the algorithm presented in [15] to generalise the sampling of emission flows including the one-loop corrections.…”

“…In particular, the anomalous dimension will be expressed in terms of variables which can be linked to the real emission contributions. Such a form is then very well suited for a Monte Carlo evolution algorithm like the one outlined in [15], and further development thereof. The approach exploited here then also allows to properly subtract collinear divergencies, and to make use of colour conservation in order to factorise softcollinear from (colour) non-trivial soft, large-angle physics, see the discussion on collinear subtractions and the ordering variable in [7].…”

“…Such a formalism has recently been adopted to make decisive statements about the accuracy of parton showers [8][9][10][11][12]. Solving the evolution equations which do resum non-global logarithms requires Monte Carlo methods [13] and has recently been extended beyond the leading-N limit [14,15], addressing a dedicated resummation algorithm before aiming at a more versatile simulation within an event generator. The same formalism also allows to analyze approaches to improve existing parton showers beyond the leading-N limit [16][17][18] in order to show which colour suppressed contributions are actually taken into account [19], highlighting the fact that amplitude level evolution will go beyond a probabilistic approach in the sense that unitarity cannot be naively employed anymore.…”