Next-to-leading order QCD calculations with parton showers II: soft singularities

TL;DR

This work extends NLO QCD calculations by integrating parton showers, with a focus on soft, wide-angle gluon emissions. It achieves this by formulating soft radiation as coherent antenna emission from the Born three-jet configuration in the Coulomb gauge, deriving real and virtual soft-subtraction terms and an exponentiation structure. The method preserves NLO accuracy when showers are added, embedding primary splittings and soft radiation into the perturbative calculation and validating the approach with a numerical thrust test. While promising, the work also outlines necessary future steps, including merging with two-jet calculations and incorporating hadronization and color flow for broader applicability.

Abstract

Programs that calculate observables in quantum chromodynamics at next-to-leading order typically generate events that consist of partons rather than hadrons -- and just a few partons at that. These programs would be much more useful if the few partons were turned into parton showers, which could be given to one of the Monte Carlo event generators to produce hadron showers. In a previous paper, we have seen how to generate parton showers related to the final state collinear singularities of the perturbative calculation for the example of e+ + e- --> 3 jets. This paper discusses the treatment of the soft singularities.

Figures (5)

• Figure 1: Primary parton splitting from Ref. I. The filled circles represent graphs for the Born amplitude and complex conjugate amplitude. Each of the partons emerging from the Born amplitude splits into two partons with a vertex, represented by the squares, that includes a Sudakov suppression factor. Each of the six daughter partons undergoes further, secondary, splittings and enters the final state as a complete shower. The secondary splittings are represented by the diamonds.
• Figure 2: Primary parton splitting with soft gluon radiation added. The graphical symbols of Fig. \ref{['fig:shower1']} are also used here. The extra gluon coming from the circles on the right and left represents the soft gluon radiated from the three jets. As with the other partons, the soft gluon undergoes secondary splittings and generates a shower, represented by the diamonds.
• Figure 3: Comparison of the NLO calculation with showers added as described in this paper and I to a pure NLO calculation using beowulfcode. I plot the ratio $R$ defined in Eq. (\ref{['Rdef']}) for the thrust distribution at thrust equal 0.86. Also shown is the ratio $R_{\rm LO}$, defined in Eq. (\ref{['RLOdef']}), in which the order $\alpha_s^{B+1}$ correction terms are omitted from the calculation. The c.m. energy is $\sqrt S = M_Z$ and the renormalization scale is chosen to be $\mu = \sqrt S/6$. These ratios are calculated for $\alpha_s(M_Z)^2 = \{0.25,1,2,3,4\}\times (0.118)^2$ and plotted versus $\alpha_s(M_Z)^2/(0.118)^2$.
• Figure 4: Comparison of the NLO calculation with showers added to a pure NLO calculation for $t = 0.71$. The notation is as in Fig. \ref{['fig:test']}.
• Figure 5: Comparison of the NLO calculation with showers added to a pure NLO calculation for $t = 0.95$. The notation is as in Fig. \ref{['fig:test']}.