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an ultraviolet attract or (or stable fixed point). In the case reduce to the constant 0(00' In a more quantitative way let us write, close to z = 0(00'
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t/I(z) = v(O(oo - z)
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d as would
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(13-20)
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Integration of (13-11) yields
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q2) -;nz =
0(00 -
-;nz)-V + ...
(13-21)
where the positive constant k depends on the precise form of t/I. The critical index v[v = -(dt/l/dz)(O(oo) > OJ characterizes the approach to the fixed value 0(00 for which the photon propagator assumes its free-field value. Correspondingly, 0(00 would describe the electromagnetic coupling at short distances as if the bare coupling were finite.
13-1-2 The Callan-Symanzik Equation
Instead of comparing the effect of changing the renormalization point, we make seemingly a step backward by returning to the bare regularized Green functions using a cutoff A m. For an appropriate choice of bare parameters we have the relation d(0(,q2,m 2) = lim d(0(0,q2,m6,A 2) (13-22)
A-+oo
The relation as we know was only established perturbatively and does not involve any multiplicative wave-function factor due to the Ward identity. Furthermore, we have (13-23) The A -+ 00 limit will always be understood, but most of the time it is not written explicitly. The crucial observation is that, due to this limiting procedure, the renormalized amplitudes depend on one dimensional parameter less than the regularized ones. To proceed, consider the irreducible function d- 1 = 0(-1(1 + w). The diagrams contributing to
a a- d- 1 ( mo
0(0,
2 q2, mo, A2)
are superficially convergent. Except for the factor gllVq2 - qllqV they correspond to the irreducible part of the Green function (Fig. 13-1)
d x e-;q'x<OI TjIl(x)P(O)
d4 y imo: \jIot/lo(Y):
QUANTUM FIELD THEORY
Figure 13-1 Diagrams for mo(iJ/iJmo)d-1.
The superficial convergence assumes current conservation. The insertion of the operator imo Sd4y: lJIt/J(y): increases by one unit the degree of one of the denominators in the Feynman integrals. However, as discussed in Chap. 8, the subtractions of internal divergences due to the insertions of molJlot/Jo in fermion self-energy subdiagrams require the introduction of a new counterterm or, equivalently, the multiplication of lfIot/Jo by ZIi/",' Ultimately
L\ (0:, -
~:) = ZIi/",mo o!o r
1(0:0, q2, m6, A2)
(13-24)
will be finite in the limit A ---+ 00. To lowest order L\ is derived from formula (7-9). It behaves as _(m 2/q2) In (_q2/m 2) for large _q2/m 2. At order n, L\ is - m2/q2 times a polynomial in In ( - q2/m 2), and therefore its asymptotic limit L\as vanishes perturbatively. We consider now a variation of m (hence also of 0:) for fixed 0:0 and A. According to (13-23),
If this has a limit as A ---+ 00 with 0: and m fixed, the only dimensionless parameter left on which it can depend is 0:. From dimensional arguments we can therefore define
f3(0:) = lim o:m !-In Z31 '"o,A = - lim o:A "a In Z3 (A , 0: 0 ) A A-->oo um A-->oo U m
(13-25)
We shall soon verify that f3(0:) exists perturbatively. For this purpose we compute the corresponding derivative of d- 1 :
A-->oo
. hm m - a d _ 1 j '" A am 0,
[a mam
+ f3(0:) -a ] d _ 1 ( 0:, 00:
I' - m -,,omo 1m
- -q2 ) m2
From (13-24) we can also express the left-hand side as
A-->oo
a 2 I1m m -;- d - 1 (0:0, q 2 ,mo, A2 ) 1 ,"o,A
A-->oo
a x mo -,,- d umo
1 ( 0:0,
2 q 2 ,mo, A2) 1 '"o,A
. omo q2 = hm (m - -) Z~ 1 L\ ( 0:, - - 2 ) A-->oo
mo am
"''''
ASYMPTOTIC BEHAVIOR
If f3(IX) is finite the same should be true for
. m omo 1 + b(lX) = hm Z= 1 ~ - A ... mo am
00 "''''
'"G,A
(13-26)
leading to the Callan-Symanzik equation
o~ + f3(IX) :IXJrl (IX, [2x Ox
+ b(IX)] L\ (IX,
!: )
(13-27)
From the fact that L\as vanishes it also follows that
a + f3(IX) OIX d -1 (IX, x) = OJ
(13-28)
Equation (13-28) implies that f3(IX) is finite perturbatively. It is sufficient to use the equation for some definite value of x. Conversely, we may derive renormalizability from a study of this equation and its extension to other Green's functions. From the three-loop computation of de Rafael and Rosner we obtain for fermionic quantum electrodynamics
f3(IX) =
h + 2n 2
21X2
121 1X4 144 n 3
+ O( 1X5)
(13-29)
In contradistinction to the function 1/1, the function f3 depends on the convention used to define the renormalized coupling IX, except for its first two terms. The above expression shows that for IX --+ +0, f3(IX) is positive. As promised before, let us now derive the leading behavior of the vacuum polarization to an arbitrary large order as a consequence of (13-28) and (13-29). We substitute in (13-28) the formal series
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