represented by single lines only but gauge bosons being Nc x N, matrices, by double
lines, incoming and outgoing. The Kronecker delta's identifying the matrix indices in the
traditional Feynman rules are directly and manifestly implemented by connecting the
single lines at the vertices, as shown in figure 1-1 below.
(k,) (i,) -gpk6jl/(k2 1)
(n,j)
S (gav(k- q)p/ +gap(P k), +g,(q P)a
P q
(1, k) P (m, i)
i i
SGauge Propagator as above without 6-s.
SJ Cubic vertex as above without 6-s.
Figure 1-1. Example of 't Hooft's double-line notation. Each line hold a single matrix
index i or j resulting in the Gauge propagator to be represented by two
such lines. This manifestly implements the Kronecker delta's present in the
propagators as shown. An incoming arrow on a single line denotes that the
index it holds refers to a row and an outgoing arrow to a column index. This
means that fermion propagators (not shown) would be represented by single
lines.
The double line notation can also be implemented for Fadeev-Popov ghosts in the
Feynman gauge. In this notation, consider a large Feynman diagram with the Kronecker
delta's manifestly implemented graphically. A general such diagram would often require