Eikonal gluon bremsstrahlung at finite $N_c$ beyond two loops

We present a general formalism for computing the matrix-element squared for the emission of soft energy-ordered gluons beyond two loops in QCD perturbation theory at finite $N_c$. Our formalism is valid in the eikonal approximation. A Mathematica program has been developed for the automated calculation of all real/virtual eikonal squared amplitudes needed at a given loop order. For the purpose of illustration, we show the explicit forms of the eikonal squared amplitudes up to the fifth-loop order. In the large-$N_c$ limit, our results coincide with those previously reported in literature.

Figure 1

The first-order correction to the simple QCD process: electron-positron annihilation into a quark and antiquark accompanied with the emission of a soft gluon $g$.

Figure 2

One-loop vertex correction to the process $e^{+}{e}^{-}\to q\overline{q}$.

Figure 3

Feynman diagram for one of the contributions to the emission amplitude of four gluons in the process $e^{+}{e}^{-}\to q\overline{q}$.

Figure 4

A diagrammatic representation of the reducible (left) and irreducible (right) contributions to the two-loops eikonal amplitude squared ${\mathcal{W}}_{12}^{\mathrm{R}\mathrm{R}}$.

Figure 5

A diagrammatic representation of the reducible and irreducible contributions to the three-loops eikonal amplitude squared ${\mathcal{W}}_{123}^{\mathrm{R}\mathrm{R}\mathrm{R}}$.

Figure 6

The term ${\mathcal{A}}_{ab}^{12}{\overline{\mathcal{A}}}_{ab}^{13}$ of the irreducible contribution (45a).

Figure 7

A diagrammatic representation of the reducible contributions to the four-loops eikonal amplitude squared.

Figure 8

A diagrammatic representation of the irreducible contributions (51) to the four-loops eikonal amplitude squared.

Figure 9

An example of Feynman diagrams, for the amplitude squared, which are not symmetric under the interchange of one or more pairs of emitters of gluons $k_2$, $k_3$, and $k_4$.

Figure 10

Eikonal Feynman rules for QCD. The “double” lines represent the radiating quark and gluon eikonal lines.