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November 7, 2013 61
1.6.6 Low-order approximation to the renormalized coupling
Let us examine the possible shape of the function F (v, s) in some more detail.
In the spirit of perturbation theory, it will be given by a series expansion like
F (v, s) = v + v 2 α 1 (s) + v 3 α 2 (s) + v 4 α 3 (s) + · · · , (1.147)
where the functions α j (s) vanish at s = 0. The beta function is then given by
β(v) = F 2(v, s)
F 1 (v, s) = v2 α 1(s) ′ + v 3 α 2(s) ′ + v 4 α 3(s) ′ + · · ·
1 + 2vα 1 (s) + 3v 2 α 2 (s) + · · ·
, (1.148)
so that we see that it must start with v 2 :
β(v) = β 0 v 2 + β 1 v 3 + β 2 v 4 + · · · (1.149)
The requirement that the beta function depend not on s governs the form of
the functions α j (s) : to low order in v we have from Eq.(1.148)
so that we can derive
β(v) = v 2 α ′ 1(s) + v 3 ( α ′ 2(s) − 2α 1 (s)α ′ 1(s) ) + · · · , (1.150)
α 1 (s) = β 0 s , α 2 (s) = (β 0 s) 2 + β 1 s , . . . (1.151)
It is easily derived that the leading term in α n (s) is (β 0 s) n .
Let us assume that the beta function is dominated by its lowest-order term,
that is, β(v) = β 0 v 2 . In that case, h(v) = −1/(β 0 v), and we find
1
w(s) = 1 v − β 0s . (1.152)
We can exchange the bare parameter v for the measured value of w at some
fixed scale s 0 , and then the running is given by
or
1
w(s) = 1
w(s 0 ) − β 0(s − s 0 ) , (1.153)
w(s) =
w(s 0 )
1 − β 0 w(s 0 )(s − s 0 ) . (1.154)
At this point we may start to distinguish between different theories. The renormalized,
physical parameter w is a priori unknown, and has to be determined
by experiment ; but the number β 0 is perfectly computable from inside the
theory 44 . The running of the coupling is therefore determined as soon as the
44 The number β 0 is a combinatorial factor with the addition of some powers of π, and simple
numbers depending on the ingredients and quantum numbers of the particles pertaining to
the theory.