Pictures Paths Particles Processes

54 November 7, 2013
C
naive 2
C
naive 3
C
naive 4
C
naive 5
C
naive 6
C
naive 7
= ¯h + 1 2¯h2 − 3 4¯h3 + O (¯h 4) ,
= −¯h 2 − 1 2¯h3 − 131
24 ¯h4 + O (¯h 5) ,
= 2¯h 3 − 2¯h 4 − 147
4 ¯h5 + O (¯h 6) ,
= −5¯h 4 + 61
2 ¯h5 + 5665
24 ¯h6 + O (¯h 7) ,
= 10¯h 5 − 295¯h 6 − 5105
4 ¯h7 + O (¯h 8) ,
= 35¯h 6 − 5195
2 ¯h7 − 47075
24 ¯h8 + O (¯h 9) ,
(1.122)
However, we see that now C 2,3,4 = E 2,3,4 no longer hold, and therefore we must
re-tune the parameters order by order in perturbation theory. In the present
case, we find up to two-loop accuracy :
µ = 1 + 1 2¯h + ¯h2 + O (¯h 3) ,
λ 3 = 1 + ¯h − 49
24¯h2 + O (¯h 3) ,
λ 4 = 1 − 3 2¯h + 5 4¯h2 + O (¯h 3) , (1.123)
and the renormalized connected Green’s functions, suitably truncated to the
correct order in ¯h, read
C 1 = − 1 2¯h + 1 24¯h2 + O (¯h 3) ,
C 2 = ¯h ,
C 3 = −¯h 2 ,
C 4 = 2¯h 3 ,
C 5 = −5¯h 4 + 3¯h 5 − 5 2¯h6 + O (¯h 7) ,
C 6 = 10¯h 5 − 45¯h 6 + 90¯h 7 + O (¯h 8) ,
C 9 = 35¯h 6 + 480¯h 7 − 2065¯h 8 + O (¯h 9) . (1.124)
The difference between the ‘naive’ and the renormalized connected Green’s functions
is quite evident. In particular C 2,3,4 are completely free of higher-order
corrections. For the other connected Green’s functions the coefficients in the
perturbation expansion tend to be smaller in absolute value than in the ‘naive’
expressions.
The above discussion is obviously only a drastically simplified example of a
phenomenological situation that is usually much more complicated. For instance,
one does not, usually, renormalize connected Green’s functions but
rather quantities extracted from scattering matrix elements, that are themselves