Pictures Paths Particles Processes

November 7, 2013 261
Here, the B n are the Bernoulli numbers, defined by their generating function as
follows :
F (x) ≡
x ex
e x − 1 = ∑ x n
B n . (10.37)
n!
n≥0
It is easily seen that B 0 = 1 and B 1 = 1/2 ; but more significantly, from the fact
that F (x) − x/2 is actually 7 even in x, we see that all B n vanish for odd n ≥ 3 ;
which again means that all odd loop corrections beyond the first order vanish
for all Green’s functions ! The receipe is even simpler than in the previous
case : replacing each vertex by −1 and each propagator by 1, all odd-loop
Green’s functions must be identically zero ; yet another powerful check on our
computations.
7 From the way it is written, this seems unlikely — but it is true.