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da,(o:,x,)
dz !/J(z)
(13-11)
ASYMPTOTIC BEHAVIOR
equivalent to the series
das(cx, X2)
I: ~ (In x2)n {[ljJ(Z) dd In z} I on.
Xl Z
(13-12)
z=d~(a,xl)
Before giving a general discussion let us recall our earlier results (8-130) for the vacuum polarization. Remember that the power of cx is given by the number of loops. To order cx 2 , 2 -as W (CX, X) = - -cx ( In X - - - cx 2 (In X - C 2) + O(CX 3) (13-13) 3n 3 4n
with C 2 a numerical constant. If CXI stands for
(13-14)
we have
CXI as d (cx, x) = -1---(-cx-r/C"-3-n)-cl- -x---(cx"""i-:-/4-:---n-;02-)1c-- x . n n -+-'-.
(13-15)
To obtain the function ljJ(CXI), we compute the derivative (%x)das(CXI, x) for x equal to one. Let us also include the result of the three-loop calculation performed by Baker and Johnson. This gives
ljJ(z) = 3n
where
+ -Z32 + -Z43 [ ~(3) 4n 3n
~(3) =
101J + 0(Z5) 96
(13-16)
is the Riemann function
1.202 ... (13-17)
The solution of the Gell-Mann and Low equation (13-10) shows that it is possible to obtain the dominant behavior of the vacuum polarization to order n, knowing d as and therefore ljJ to lower orders. Let us illustrate this remark by restricting ljJ to the two-loop value ljJ(z) = z2/3n + z3/4n 2 + ... and deriving the dominant cx 3 (ln x term in d. From Eq. (13-11), In x =
d",(a,x)
dz ljJ(z) = J(das(cx, x)) - J(cxd 3n
(13-18)
J(z)
9 constant - ~ - 4ln z + O(z)
Upon inversion this yields
das(cx x) = CXI , 1 - (cxr/3n) In x - (cxI/4n 2) In x - (cxi/24n 3 )(ln X)2
+ O(cxr In x)
(13-19)
1 + 5cx/9n
+ C 2cx 2/4n2
= cx - -
5 2 cx + ... 9n
QUANTUM FIELD THEORY
Consequently, we have indeed found the dominant behavior to order a 3 . More modestly, with only the first term in ljJ we would have found the cancellation of a 2 (ln x terms in the two-loop computation of W, a result which cost us considerable effort in Chap. 8. Note, however, that it would have been insufficient to obtain the coefficient of a 2 In x. If al is substituted as a function of a in (13-19), we recover non leading contributions to a given order. At any rate, the conclusion to be drawn from (13-19) is that as - q2 goes to infinity the three-loop contribution to W behaves as - (a 3124n 3 ) [In ( - q2 Im 2)J2. The renormalization group therefore allows us to derive new results, while it seemed at first that we were only stating trivialities. We will show below that the numerical expressions obtained so far are sufficient to predict all terms of the form an [In ( - q2 1m 2)] n- 1. The function ljJ is only known as a power series around the origin. We have no guarantee about its convergence. On the contrary we suspect such a series to be at best asymptotic (Sec. 9-4). We recall that its range of validity requires a In (- q21m 2) -4 0 when - q21m 2 -4 00. This explains why the existence of a pole in d in the vicinity of the unphysical euclidean point
is rather doubtful. Such a pole, sometimes referred to as the Landau ghost, occurs when keeping only the first-order contribution in W. Since this singularity is clearly in the asymptotic region, a serious investigation would require the complete expression of das, all of its terms being of comparable size. It is therefore hazardous to draw any conclusions on the inconsistency of quantum electrodynamics at this stage. On the other hand, assuming ljJ to be meaningful, it is possible to speculate on the theory as a whole by formulating precise hypotheses on this function. It has a universal character independent of normalization conventions as a consequence of its very definition. Let us study a few possibilities. As suggested by the first terms in its expansion, assume that ljJ(z) is positive in an interval 0 < z < aoo To be consistent we suppose that al is included in this interval. If aoo is infinite, i.e., if ljJ(z) only vanishes at the origin, it is necessary that the integral JOO dzN(z) diverges at its upper limit in such a way that daS(a, x) goes to infinity as x -4 00. Otherwise das(a, x) becomes infinite for a nonphysical value q2 = - m2 exp [J~ dzN(z)] and the same is likely to be true for the complete function d. In other words, we have a genuine Landau ghost. Clearly the first few terms in ljJ cannot give a proper indication to test this hypothesis. The second possibility is that ljJ(z) has a finite positive zero aoo For 0 < al < aoo the positivity of ljJ(z) implies that das(x) is an increasing function and we require Ja dzN(z) to be divergent. Such is the case if ljJ(z) has a simple zero. Under these conditions das(a, x) tends to aoo when x -400. In fact, since ljJ(z) decreases in the vicinity of this point, d as tends to aoo no matter whether al is smaller or larger than a oo This property is characterized by saying that a", is
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