EFFECTIVE FIELD THEORIES FOR VECTOR PARTICLES AND ...

24 power counting and regularization
external momenta. Therefore, one can simply replace the integral H by
its IR-regularized part
H IR = H − R = I , (3.9)
considering that the part R can be compensated by redefined parameters
of the most general Lagrangian. Note that the splitting might introduce
additional divergences in I and R which cancel in the sum I + R. In
general, infinities are neglected according to the so-called ̃MS renormalization
scheme, i.e. terms proportional to the infinite quantity
λ = 1
16π 2 { 1
n − 4 − 1 2 [ln(4π) + Γ′ (1) + 1]} (3.10)
are set to zero, arguing that they also can be absorbed in counter-terms of
the most general Lagrangian. The part R in equation (3.9) is also denoted
subtraction term since the regularized integral is obtained by subtracting
R from the original integral. Furthermore, the regular part R satisfies
the Ward identities separately from I and, hence, the IR regularization
preserves the symmetries of the theory.
In the original approach of Becher and Leutwyler [17], the crucial step
is to calculate the singular part I directly in order to obtain the regularized
integral, see also [68]. This turns out to be difficult in generalized
situations. However, the subtraction terms can also be obtained order
by order by expanding the integrand in equation (3.5) directly in small
Lorentz-invariant quantities, say M 2 and p 2 − m 2 , and interchanging
the series and the integration. It has been shown that this procedure is
equivalent to the original approach order by order, see [18] for details. In
the following, this approach is termed reformulated IR regularization.
In practical calculations, the reformulated procedure provides an
easier method of finding the renormalized version of the results. This
concerns the integration as well as the identification of R, e.g. in two-loop
calculations. After a standard Passarino-Veltman reduction [72, 73] in n
space-time dimensions, the scalar integrals containing only pion masses
and small momenta, such as the external photon momentum, are kept,
whereas integrals containing only large masses and external momenta
are discarded. This implies that diagrams with loops containing only
heavy degrees of freedom can be discarded directly. Next, integrals
which contain both scales are calculated in n dimensions as explained
above up to a sufficient order in the small invariant quantities. Finally,
the subtraction terms are obtained by the expansion around n = 4,
neglecting divergences according to ̃MS scheme. Note that even after