EFFECTIVE FIELD THEORIES FOR VECTOR PARTICLES AND ...

22 power counting and regularization
and thus the splitting depends on the used renormalization scheme. In
conclusion, using an appropriate renormalization scheme enables us to
restore a consistent power counting including heavy degrees of freedom.
The extended power counting rules, which allow for assigning a chiral
order to each diagram, read as follows. First, the list of small quantities,
collectively denoted as q, needs to be extended by the expression K 2 −
MR 2 = O(q1 ) if K is a large momentum since the resonance is regarded
as nearly on-shell, K 2 ≈ MR 2 . Next, it is necessary to investigate every
possible flux of the external momenta through each diagram. For each
given flux the order of vertices and propagators are determined and
summed up as detailed below. Finally, the lowest order resulting from
the various flux assignments is defined to be the chiral order of the
diagram. The order of the vertices can be read off the corresponding
Feynman rules taking into account the previously assigned flux of large
external momentum. Additionally, one considers that the pion mass
counts as O(q 1 ), that the vector meson masses count as O(q 0 ), and
that each loop integration counts as O(q 4 ), as usual. The order of the
propagators for small and large momenta can be read off the following
table:
Momentum π ρ or ω
Small O(q −2 ) O(q 0 )
Large O(q 0 ) O(q −1 )
Table 3.1
This can be motivated by the following approximative considerations of
the typical pole structure of a propagator:
1
k 2 − M ≈ 1
2 M = 2 O(q−2 ) ,
1
K 2 − M ≈ 1
2 K = 2 O(q0 ) ,
1
≈ 1
(3.2)
= O(q 0 ) ,
k 2 − Mρ
2 Mρ
2
1
= O(q −1 ) (see text above) ,
K 2 − Mρ
2
where k represents a small momentum and K a momentum with at least
one large component, say the zeroth, corresponding to the large mass of
a rho meson.