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November 7, 2013 53
Order by order, the parameters keep getting updated, but in the overall picture
they are just bookkeeping devices that allow one to go from measurements to
predictions of the more physically interesting connected Green’s functions. It
should not come as a surprise that in the measurement-parameter-prediction
protocol, a higher-order correction in the parameters due to an improved measurement
expression is cancelled again, to some extent, in the prediction. In
fact, for certain classes of theories, which are called renormalizable, these cancellations
may be quite extreme.
1.6.2 The renormalization program : an example
As an example of the renormalization program, we shall investigate ϕ 3/4 theory.
To order O (¯h) in perturbation theory, the first few connected Green’s functions
are given by
(
C 1 = ¯h − λ )
3
2µ 3 + O (¯h 2) ,
( ( 1
C 2 = ¯h + ¯h
µ) 2 − λ 4
2µ 3 + λ 3 2 )
µ 4 + O (¯h 3) ,
(
C 3 = ¯h 2 − λ ) (
3
µ 3 + ¯h 3 − 4λ 3 2
µ 6 + 7λ )
3λ 4
µ 5 + O (¯h 4) ,
(
C 4 = ¯h 3 − λ 4
µ 4 + 3λ 3 2 ) (
µ 5 + ¯h 4 4 24λ3
µ 8 + 7λ 4 2
2µ 6 − 59λ 3 2 )
λ 4
2µ 7 + O (¯h 5) ,
(
C 5 = ¯h 4 10λ3 λ 4
µ 6 − 15λ 3 3 )
µ 7
(
+ ¯h 5 3 605λ4 λ 3
2µ 9 − 192λ 3 5
µ 10 − 80λ 4 2 )
λ 3
µ 8 + O (¯h 6) , (1.119)
and of course the next-order corrections and connected Green’s functions are
easily computable. Let us assume that the experimental values of the connected
Green’s functions C 2,3,4 have been measured, with negligible experimental error
for simplicity. We shall denote these values by E 2,3,4 , respectively. For purposes
of illustration, we shall assume that these values are
E 2 = ¯h , E 3 = −¯h 2 , E 4 = 2¯h 3 . (1.120)
In lowest order of perturbation theory, we can then find the action’s parameters
to be
µ = 1 , λ 3 = 1 , λ 4 = 1 . (1.121)
If this were all, we could then compute the connected Green’s functions. This
‘naive’ treatment would give the following results up to two loops :
C 1
naive
= − 1 2¯h + 1
24¯h2 + O (¯h 3) ,