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The Effective Coupling in QCD (1978) |
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(1) Gauge invariance formally requires 
, but the limit 
may be taken without damage. 
is governed by, for example, the spins of 
and 
. By changing one essentially changes the process under consideration.
(2) Note that the construction for 
works only to 
. In higher orders, it is necessary to specify the coupling constant by giving its size at 
, where 
is a fixed reference mass. This procedure is many cases more satisfactory in general, since it avoids the exponential sensitivity of .
(3) The sensitivity is to be expected because
(4) 
corresponds to the well-known case 
. It is conventional in QED to define 
from the low-energy limit of 
, that is, at 
. In this case the first formula in (8) fails; the assumption 
is incorrect. When the exact formula is used the logarithm for becomes 
[2].
(5) To 
the inclusion of the 
term in 
results simply in the addition of
to the denominator in eq. (8). To obtain the exact result one must solve a complicated transcendental equation for 
in terms of 
.
(6) The results for some of the necessary integrals are given in [5]. For 
, 
is added to the constant term.
