Inclusion of the QCD next-to-leading order corrections in the quark-gluon Monte Carlo shower

A methodology of including QCD next-to-leading-order corrections in the quark-gluon Monte Carlo shower is outlined. The work concentrates on two issues: (i) constructing the leading-order parton-shower Monte Carlo from scratch, such that it rigorously extends collinear factorization into the exclusive (fully unintegrated) one which we call the Monte Carlo factorization scheme; (ii) introducing next-to-leading-order (NLO) corrections to the hard process in this new environment. The presented solution is designed to be extended to the full NLO-level Monte Carlo, including NLO corrections not only in the hard process but in the whole shower. The issue of the difference between the factorization scheme implemented in the Monte Carlo (MC) solution and the standard $\overline{MS}$ scheme is addressed. The principal MC implementation is designed for the electroweak boson production process at the LHC, but in order to discuss universality (process independence), the deep inelastic lepton-hadron scaterring is also brought into the MC framework.

Figure 1

The EGMPR [4] factorization for EW boson production. Example 2PI kernels for a quark in the forward hemisphere $K_{0F}$, an antiquark in the backward hemisphere $K_{0B}$, and the hard-process part $C_0$ are delimited by ellipses. The lower figure highlights the use of cut diagram notation by indicating the amplitude in blue (left parts of the ladders) and the complex conjugated amplitude in red (right parts of the ladders).

Figure 2

The EGMPR factorization for EW boson production; the LO ladder with the projection operators ${\mathbb{P}}^{\prime }$ inserted between the LO kernels.

Figure 3

The Sudakov plane of single-gluon emission in DY for angular ordering.

Figure 4

Kinematics of one-gluon emission in the Breit frame.

Figure 5

The logarithmic Sudakov plane for single-gluon emission in the DIS process.

Figure 6

The Sudakov plane of the LO MC for DIS. The shaded area denotes the integration domain for the Sudakov form factor $S$ of the LO MC. Rapidity equal to $\Xi$ marks the boundary between ISR and FSR.