dg goo - g in .NET framework

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and g(A) will tend to goo as an inverse power of A. A similar analysis can be carried out for a multiple zero. On the other hand, nothing spectacular is expected
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Figure 13-5 Ultraviolet fixed point.
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656 QUANTUM FIELD THEORY
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from y(g) in the vicinity of gOO" If Yoo = y(goo), then up to a A-independent factor
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In a general situation, attractive and repulsive fixed points appear successively along the g axis. According to the initial value g(l), the effective coupling constant g(A) will tend to the nearest ultraviolet attractive fixed point as A ~ 00.
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Ultraviolet unstable points, (g - goo)f3(g) > 0, are infrared attractors and correspond to limits of g(A) when A--> O. This is of interest when we study, for a massless theory, the limit of large distances or very small momenta with respect to an ultraviolet cutoff. This is precisely the aim of the theory of critical phenomena where m = 0 corresponds to the critical temperature and we investigate the long-range tail of correlations.
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Returning to our original problem we conclude from the existence of a nearby ultraviolet fixed point that for large A the asymptotic behavior of amplitudes is of the form
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(13-87)
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Scale in variance has been recovered, corresponding to a specific value goo of the coupling and with an effective dimension for the field
(13-88)
The Green functions with a mass insertion Ll obey (-IPk' O!k +f3(g):g +{2-n[1 +y(g)] +y,,(g)})rrl,(O;Pi;g)=O with y,,(g) = 0 In Z",2/0 In A [see Eq. (13-75)]' As long as y,,(goo) < 2, the perturbative hypothesis of identifying the asymptotic regime with the massless theory is consistent. Otherwise the mass insertion would, in effect, correspond to a hard operator and the theory should be reformulated. See Sec. 13-3-3 for additional remarks on this point.
This beautiful reasoning, essentially due to Wilson, shows how the renormalization group has been an instrument in recovering a nontrivial scaling behavior. The crux is now to find whether P(g) has such zeros. Since, however, the origin is always such a zero, as P(g) vanishes in the absence of interactions, it is of special interest to assert perturbatively the nature of this fixed point.
Exercises
(a) Discuss the effect of a multiple zero on the asymptotic scaling law and find the corrections to the
dominant behavior.
(b) Show that the positive measure in the Lehman-Kallen representation for the two-point function implies y :2: 0 and that Yoo = 0 implies a free theory as noted by Parisi and Callan and Gross.
ASYMPTOTIC BEHAVIOR
13-3-2 Asymptotic Freedom
We have given examples of the computation of [3(g) for small g in the case of a self-coupled scalar field and abelian or nonabelian gauge fields. The results are represented in Fig. 13-6. For electrodynamics we have plotted the quantity 4n[3(rx)/2e [with [3(rx) given by Eq. (13-29)J as a function of e. If the origin is an ultraviolet fixed point we say that the theory is asymptotically free. Among the examples exhibited here, only nonabelian gauge fields with a small number of fermions possess this property. Superficially <p4 theory would also be in this case for g < O. But it is likely to be unstable for negative g. Since asymptotic freedom would mean that at least for the large momentum regime radiative corrections are calculable perturbatively, we would then- find an unbounded effective potential. We conclude with Coleman that for g < 0 the theory is unstable. An exhaustive examination by Coleman and Gross establishes the following result. No renormalizable field theory is asymptotically free in four dimensions, except if it contains nona bel ian gauge fields.
We outline the derivation of this important result.
(a) Scalar theory The ({J4 interaction is generalized to an arbitrary set of interacting scalar fields in the form
with gijkl totally symmetric. To order h,
fJijkl = -1.
d-1.
~ gijkl(-1.) =
L [gilmn(-1.)gjkmn(-1.) + permutations]
with A > 0, Assuming the quartic form gijkl({Ji({Jj({Jk({J1 positive for stability, we derive that fJiiii> 0,
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