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where and obtain the following equation:
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b2 = 2n 2
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d n 2 - Pn+l(x)=L(Q-1)pq{x)b n- Q +1 dl n x Q=2
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The solution is Pi (x)
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b1 -2Inx+Cl
Pn(x)
(b; )n-2
(Inn ~n~l
(13-32)
+ O(ln x n- 2)
n> 2
QUANTUM FIELD TIIEORY
This shows that computations up to the two-loop order yield hI and h2' which in turn determine the leading behavior to order n. We may reorganize the expansion of d;;.1 by summing successively leading logarithms, sub leading ones, ... :
d;;' I (a, x)
= a-I
(1 - 3: In x)
+ }n In (1
- 3: In x)
+ C I + ...
(13-33)
Unfortunately this method does not enable us to reach d;;.l(a, x) for large x. For instance, if we were to keep leading logarithms we would recover the unphysical Landau ghost by setting 1 - (a/3n) In x equal to zero. For this value the second term is logarithmically infinite, violating the assumption that it" is subdominant. Stated differently, the reorganization implied by (13-33) is only useful when [a In x [ 1. To find the true behavior for large x requires to sum up all logarithms or, equivalently, to return to the original equations (13-27) and
(13-28).
It is interesting to understand diagrammatically why it is sufficient to know b1 and b2 alone in
order to obtain the coefficient of the leading logarithm to any order. We recall an observation made in Chap. 8, according to which when summing gauge-invariant classes of diagrams the leading power of In x is equal to the number of internal fermion loops. The reader will find it instructive to derive this property by the device of introducing N-degenerate species of fermions and studying the dependence on N. Diagrams with the largest number of fermion loops at a given order in IX are of the type shown in Fig. 13-2. The resulting structure shows why a two-loop calculation yields the coefficient of the dominant term to order n as proportional to M -2b 2 (ln x)n- 1. From (13-25) it follows that the constant Z3 satisfies the equation
~ IX [ -2 Y oy + P(IX)
~OIX
1)]Z3(Y, IX) = 0
(13-34)
=-;;;z
Therefore it is possible to study the limit of infinite cutoff Y -> 00, provided we are willing to make some hypothesis on the function P(IX).
The Gell-Mann and Low function !/J and the Callan-Symanzik coefficient
p(a) are, of course, related. Take, for instance, the derivative of the expression
crossed
Figure 13-2 Diagrams with the maximal number (n - 1) of fermion loops contributing to vacuum polarization at order n.
ASYMPTOTIC BEHAVIOR
(13-11) with respect to Xz, and substitute in (13-28). We obtain 1/1 [das(a, x)]
2. p(a) aa das(a, x)
(13-35)
which may be specialized to some value of x. A possible choice is x = 1(qZ = - m Z), with das(a, 1) = al (a) = a - 5a z/9n + ... , and
I/I(al) =
2. p(a) da
(13-36)
This also shows that the zeros of the two functions are related. If ac is such that p(ac ) = 0, then 1/1 vanishes for a oo = al (a c )' In particular, when this value is an ultraviolet fixed point, das tends to a oo as x goes to infinity. Furthermore, from (13-34) Z3 is finite and a oo plays the role of a finite bare square charge. Adler has given a thorough discussion of the possibility that ac might be equal to the observed physical fine structure constant. Unfortunately, at present we lack a definite nonperturbative procedure to evaluate these functions, so that the whole matter remains speculative.
Derive for the other Green functions of electrodynamics the analogous Callan-Symanzik equations.
13-2 BROKEN SCALE INV ARIANCE
As realized some time after the early investigations of Gell-Mann and Low, a more general point of view is to investigate the short-distance behavior of Green functions in a renormalizable theory when all relative distances are space-like and tend to zero simultaneously. This question may seem purely theoretical since it involves amplitudes far off-shell. Luckily this is a wrong impression. Indirect means, such as deep inelastic lepton scattering on hadronic targets, enable one to probe short-distance interactions. The results of such experiments, anticipated by theoretical considerations of Bjorken and Feynman, partly motivated these investigations by Wilson, Symanzik, and Callan. The expectation that at large momenta when masses become negligible the theory becomes scale invariant is in fact too naive. The asymptotic behavior is given by the corresponding massless theory. Renormalization imposes the choice of an arbitrary energy scale as discussed in Chap. 8. This scale spoils dimensional analysis. Its very arbitrariness is in fact what saves the day. A change of scale may be absorbed into a modification of the coupling constants. The corresponding flow is governed by functions similar to the p coefficient encountered above, and the renormalization group transformations appear therefore as a substitute to naive dimensional analysis. Ultraviolet fixed points in this flow (if they exist) will attract the coupling constants as A increases to infinity. This leads to the restoration of short-distance scale invariance for specific values of the couplings largely independent of the
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