In the perturbative limit, the cross section of QCD scattering processes factorizes into hard and soft operators, A and S respectively. These operators act on and evolve states in the space of color charges, and depend on the resolution scale E (which, in the simplest cases, describes the energy of the emitted gluons). Intuitively, the soft operator S describes the hadronization of the incoming hard partons into hadrons, and the hard operator A encapsulates the emission of (non-collinear) soft gluons off the hard partons.
In this project, we aim to better understand the physical underpinnings of QCD cross sections described above. In particular, we will be taking two complementary approaches:
Firstly, we seek to unpack the term Tr(AS) of the cross section: In this term, one contracts diagrams describing the hadronization of the hard partons with soft processes encapsulating the eikonal interactions of the partons. The components arising from S are expressed in a multiplet basis, where each component in the tensor product of various disjoint multiplets corresponds to a particular post-collision hadronic state into which we force the corresponding hadrons. These multiplets correspond to the irreducible representations of SU(Nc), the gauge group of QCD where Nc is the number of colors, and thus impart the corresponding symmetries on the final states of the collision. On the other hand, the objects arising from A are traditionally expressed in the color flow “basis” (the color flow basis is overcomplete and thus technically not a basis); the reason for this is numerical in nature, as the color flow basis allows for a fairly straight-forward implementation of Monte Carlo methods. From a mathematics perspective, when contracting objects of these two bases (the color flow and multiplet bases), it is not a priori clear which of these objects will have a non-trivial overlap. In this first part of the project, we strive to formulate a list of criteria describing the contraction behaviour of elements of these two bases, as such knowledge will simplify subsequent numerical implications and provide insight in the structure of hadronization models.
The second object of interest in this project is the soft anomalous dimension matrix. Recent work has shed light on the two-loop calculation of the anomalous dimension matrix in the color flow basis. The problem that arises here is similar to that already alluded to in the previous paragraph: it is not a priori clear which objects in the color flow basis, when contracted with certain objects in the multiplet basis, will yield a non-zero output. To phrase this in physics terms, it is not clear which soft processes (encapsulated in the soft anomalous dimension matrix) give rise to which singlet states after hadronization. A reformulation of the soft anomalous dimension matrix in the multiplet basis would forego this problem as the non-zero overlap between two states is trivial by the mutual transversality of those states.
Research stays: 11 July 2022 - 10 November 2022
|Judith Alcock-Zeilinger||University of Tübingen|