Abstract
It has been suggested that a colour-entanglement effect exists in the Drell-Yan cross section for the ‘double T-odd’ contributions at low transverse momentum \bm{Q_\st}, rendering the colour structure different from that predicted by the usual factorisation formula . These T-odd contributions can come from the Boer-Mulders or Sivers transverse momentum dependent distribution functions. The different colour structure should be visible already at the lowest possible order that gives a contribution to the double Boer-Mulders (dBM) or double Sivers (dS) effect, that is at the level of two gluon exchanges. To discriminate between the different predictions, we compute the leading-power contribution to the low-\bm{Q_\st} dBM cross section at the two-gluon exchange order in the context of a spectator model. The computation is performed using a method of regions analysis with Collins subtraction terms implemented. The results conform with the predictions of the factorisation formula. In the cancellation of the colour entanglement, diagrams containing the three-gluon vertex are essential. Furthermore, the Glauber region turns out to play an important role – in fact, it is possible to assign the full contribution to the dBM cross section at the given order to the region in which the two gluons have Glauber scaling. A similar disentanglement of colour is found for the dS effect.
Highlights
Tors containing TMDs [9,10,11]
In [1] it was derived how the colour entanglement resulted in an additional colour factor, which reduces the azimuthal cos(2φ) asymmetry that arises from the double BM effect [25] and even changes its overall sign
As a by-product we will see that the double BM (dBM) effect at this order can be entirely ascribed to the region in which both exchanged gluons have Glauber scaling, the fact that the effect is correctly described by the factorisation formula with only TMDs and no explicit Glauber function implies that these Glauber effects can be absorbed into the TMDs
Summary
In this paper we focus on DY scattering, producing a virtual photon (or Z boson) with momentum q, which in turn decays into a charged lepton-antilepton pair with momenta l and l. An entangled colour structure contributing to the dBM term for example arises in the graph in figure 2 where there is a one-gluon exchange between each correlator and the active parton coming from the other side. Does this type of entanglement survive after summing over all relevant graphs to obtain factorisation?. In order to answer this question, we will perform an explicit factorisation calculation To this end, we will use a spectator model that we consider rich enough in structure to settle the issue – in particular, the colour factors involved are the same as those appearing in a full QCD calculation. Before we introduce the model and present the calculation, we will remind the reader of a few key steps in the derivation of factorisation
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