The One-loop QCD Corrections for $γ^* to Q\bar Q q\bar q$

Abstract

We calculate the one-loop QCD corrections for the decay of an off-shell vector boson with vector couplings into two pairs of quarks of equal or unequal flavours keeping all orders in the number of colours. These matrix elements are relevant for the calculation of the next-to-leading order ${\cal O}(α_s^3)$ corrections to four jet production in electron-positron annihilation, the production of a gauge boson accompanied by two jets in hadron-hadron collisions and three jet production in deep inelastic scattering. We use standard techniques for computing the interference of one-loop and tree level Feynman diagrams, but organise the results in terms of combinations of scalar loop integrals that are finite in the limit of vanishing Gram determinants and are therefore numerically stable.

Figures (2)

• Figure 1: The classes of Feynman diagrams relevant for the different colour structures. The solid circle indicates the possible positions for attaching the off-shell photon to the quark-antiquark pair $i,j$. Group (a) contributes to $A^{(1)}_A(i,j)$, (b) to $A^{(1)}_B(i,j)$ and group (c) to the leading colour amplitude $A^{(1)}_C(i,j)$. Diagrams with self-energy corrections on the external lines are zero in dimensional regularisation and have been omitted.
• Figure 2: The finite functions for the triply massive triangle graph with $s_{1234} = 1$ and $s_{12} = 0.2$ as a function of $\Delta_3/\Delta_3^{{\rm max}}$ where $\Delta_3^{{\rm max}} = -(s_{1234}-s_{12})^2$. The dashed lines show the approximate form for the function in the limit $\Delta_3 \to 0$, retaining only the first term of the Taylor expansion.