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Figure 13-6 The function fJ(g) near the origin: (a) self-coupled scalar field; (b) nonabelian gauge field with HC> 4Tf ; (c) nonabelian gauge field with HC < 4Tf , electrodynamics C = 0, Tf = 1, or Yukawa coupling,
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QUANTUM FIELD THEORY
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(b) Yukawa interaction
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The interacting lagrangian is of the form
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The positivity of fJiiii may be spoiled, but all Yukawa couplings cannot vanish asymptotically. Indeed, if
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l=Ak+iYsBk
then
Since abelian gauge fields are not asymptotically free either, the only possibility that remains is to include nonabelian gauge fields. If Yang-Mills fields are coupled to fermions it is necessary for a simple gauge group that the condition llC > 4Tf be fulfilled. If boson fields are also present the situation is more involved. First these fields must not be too numerous to keep fJ(g) < 0, where g is the gauge coupling. Furthermore, these fields have their own self-coupling. Assuming, for instance, that there exists a unique scalar field belonging to the adjoint representation, its self-coupling g" will obey to lowest order an equation of the type
with A > 0. The quantity a = g,,/g2 will then behave according to
d .l. d.l. a(.l.)
= Aa (.l.)
+ Ba(.l.) + C
Furthermore, we have to assume that g" is at least of order g2; otherwise the positive term Aa2 would be dominant and ruin asymptotic freedom. The right-hand side admits two roots al and a2 of order unity provided B2 > 4AC. This requires a delicate balance between the number of fields, as added fermions tend to increase the quantity B. If this is satisfied, then g,,(.l.) will behave as ag 2(.l.) when.l. ..... 00, with a one of the roots of the equation, preserving therefore the required property. The general situation with several Bose fields is even more intricate and requires a case-by-case detailed study. The introduction of such Bose fields may be motivated by the desire of generating mass terms through the mechanism of spontaneous symmetry breaking. In practice it is difficult to combine asymptotic freedom and symmetry breaking. By the introduction of a Yukawa interaction gliiifJ<P we n:ay at. best ac~ieve. an unstable situation along an a.tttactive line in the plane (g, gliiifJ<P). For a dISCUSSIOn of thIS pomt see the lectures of Gross quoted m the notes.
The conclusions derived from the calculations of the first few terms near
= 0 rely, of course, on the assumption that the perturbative series is asymptotic.
When discussing large orders (Chap. 9) we have seen that this is the best that can be hoped for. The complete series is likely to be divergent with an essential singularity at the origin. Needless to say, to find the behavior of f3(g) away from the origin remains a challenge for the future. The large momentum behavior of an asymptotically free theory becomes calculable provided the origin is the nearest fixed point. This is one reason why it is so favored by theorists. The name asymptotic freedom is, however, slightly misleading, since even then the scaling behavior departs from the free-field case by logarithmic factors. With
(13-89)
ASYMPTOTIC BEHAVIOR
we obtain, for g(A),
g (A) = 2b In A + 0
[In (In (In A
(13-90)
For most of the interesting operators the function y(g) will be of order gZ:
y(g) =
cl + 0(g4)
c dg']
~ (2bg In A)"/zb
(13-91)
Therefore the scaling factor will contain logarithms
z(A) ~ exp [ -
r9(A)
(13-92)
A Green function involving the operator Oi will then behave as (2bg Z In A)",/2b, apart from the canonical power of A. The existence of asymptotically free theories has far-reaching consequences for the purpose of building models of strong interactions. They enable us to reconcile two seemingly contradictory aspects. At low and medium energies the interactions are indeed strong and complex with numerous resonances. An approximate flavor symmetry according to SU(3) or SU(4) gives a qualitative description of hadrons as composite bound states of quarks. Every attempt at isolating these constituents has failed so far. At higher energies, however, quarks have much weaker interactions up to the point where they seem to act as free particles. The relevant kinematical domain reached by the experimental constraints is the one of light-like separations. We shall see in Secs. 13-4 and 13-5 that it is, however, possible to extend the short-distance expansion discussed so far to this region. It would then be possible to explain this paradoxical behavior by assuming an underlying asymptotically free-field theory. The catastrophical infrared singularities would be responsible for permanent confinement of quarks. These attractive speculations require necessarily such a model to include nonabelian gauge fields coupled to an unobserved (and perhaps unobservable) color degree of freedom.
The generalization to several couplings and to the corresponding multidimensional flow uncovers a great richness of phenomena such as stable fixed points, limiting cycles, ergodic behavior, etc. The difficulty of obtaining reliable information on this flow away from the origin somehow limits the interest of such a study.
13-3-3 Mass Corrections We return to the question of the consistency of the massless asymptotic theory. Nonleading powers in the large-momentum behavior of perturbative amplitudes have been neglected. They reflect the presence of terms in the lagrangian with dimension less than four and corresponding couplings with a positive dimension in energy. Typically these are the mas's terms. We have already mentioned that we can define anomalous dimensions for composite operators Oi such as <pz or !iJ!/J, thanks to their multiplicative renormalizability. Assume to simplify that the corresponding renormalization matrix is diagonal and of the form Zi(A/m, go).
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