we present here the basis of 't Hooft's results and notions which became standard in this
field.
't Hooft's arguments work equally well with pure U(NA) gauge theories as 'l. v do
with U(NA) gauge theories coupled to quarks or even scalar matrix field theories with
global U(NA) symmetry. 't Hooft presented his arguments for a theory coupled to quarks
with Lagrangian given by
L = Tr ( GP" 7 (,,D, + Tm(r,)) 4r with (1-10)
D rY = r + gA rY (1-11)
where the index r = 1,2, 3 labelled the quark family:
I1- p; 2 -n; 3 -A, (112)
each one being a vector which the fundamental representation of U(NA) can act on; i.e.
the boldface type 4, indicates a column-vector (and the 4, a line-vector) which the matrix
indices of A. can act on. The covariant derivative in Eqn. (1-11) for example contains the
matrix multiplication of A. with 4r. The G are as in Eqn. (1-7).
The Feynman-rules are obtained in the usual manner and l,. v are nicely assembled
with figures in 't Hooft's original article. To illustrate his so called double-line notation
we consider the pure Gauge propagator and cubic vertex, as shown in top two pictures in
figure 1-1. The lower two pictures in that same figure show the same obi. --t in the double
line notation. This notation is essentially a convenient way of organizing and keeping track
of the color group U(NA). Since A, is in the fundamental representation of U(NA), then
there are N2 degrees of freedom associated with it, represented by the matrix elements of
the A. with A. = -At, the d .-.-, r both transposing the matrix and taking the conjugate
of all elements. The vector and matrix indices i, j of the fundamental representation,
going from 1 to NA, are denoted by an arrowed single line, incoming arrow for row vectors
and outgoing for column vectors. It is then clear that fermions being NA-vectors will be