where
F(x) = 2f,(x) + Nfff(x) + Nf, (x) (3-7)
x(1 x) 2 for i = g gluonss)
fi(x) = < x(1 x) for i = s (scalars) (3-8)
1/2 x(1 x) for i = f fermionss).
The first two terms in the I sum produce a ln(1/m) divergence and we notice the familiar
entanglement of ultraviolet (a -- 0) and infrared (m -- 0) divergences [17]. It has been
explained how these divergences disentangle [6] and we will discuss this further in the
next section. In (3-6) Nf counts the total number of fermionic states, so, for example, a
single Dirac fermion in 4 space-time dimensions has Nf = 4. We see that Supersymmetry,
Nf = Nb = 2 + Ns, kills the I dependent term in the summand. If Nf = 8 as well, the wave
function contribution to coupling renormalization (apart from the entangled divergences)
vanishes. This is the particle content of A = 4 SUSY Yang-Mills theory.
3.2 One-Loop Gluon Cubic Vertex: Internal Gluons
Now we turn to the contribution of the proper vertex to coupling renormalization.
The proper one loop correction to the cubic vertex is represented by a Feynman triangle
graph appearing in the worldsheet as shown in Fig. 3-2. With the external particles of
Fig. 3-2 restricted to be gluons (vector bosons) the one loop renormalization of the gauge
coupling requires calculating the triangle graph for the different particles of the theory
running around the loop. In the following it will be useful to employ the "complex i -i-
x^ = x1 + ix2 and x' = x' ix2 for the first two components of any transverse vector x,
and as the name of the section -ii-.- -1 we consider first only gluons running around the
loop.