QUANTUM FIELD THEORY in VS .NET

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QUANTUM FIELD THEORY
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We may now reexamine the semiclassical picture sketched in Sec. 12-1-3 and look for solutions of the equations of motion for a theory involving gauge fields and scalar fields. Nontrivial static solutions of finite energy do not exist for gauge fields or scalar fields alone. However, theories involving both scalar and gauge fields may possess interesting classical solutions in three space dimensions. For a static solution of finite energy, the fields must tend at spatial infinity toward one of the lowest energy configurations. Otherwise the energy density would differ from zero by a finite amount in an infinite domain. A possible way to insure stability of a nontrivial solution is the existence of a set of degenerate vacua. We may then assume that the fields tend to different vacuum configurations in different spatial directions at infinity. The solution, if it exists, will be topologically stable if it maps in a nontrivial way the S2 sphere at spatial infinity onto the manifold of possible vacua, i.e., the coset space GjH. A sufficient condition is that the symmetry is spontaneously broken and that the homotopy group n2(GjH) is nontrivial. For definiteness, let us consider the Georgi-Glashow model, which is a gauge theory of symmetry group G = SO(3), where a triplet of scalar fields is coupled to the triplet of gauge fields. The symmetry is spontaneously broken into U(I) = SO(2). If the remaining massless gauge field is regarded as the electromagnetic field, we have a model of quantum electrodynamics based on the group SO(3). With this in mind, we call the coupling constant e in the sequel. Mathematicians teach us that n2[SO(3)jU(I)] = Z, the group of integers, which means that the solutions are characterized by an integer topological charge n. 't Hooft and Polyakov have studied the n = 1 solution. This corresponds to a boundary condition such that </>a points along the normal to the S2 sphere at infinity [identity map from S2 onto
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where v is the vacuum expectation value of the real field </>. If we choose the gauge that D</> vanishes asymptotically, leads to
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It is then possible to show the existence of regular solutions </>a(x), Aa(x) satisfying these boundary
conditions. The topological invariant n may be interpreted as a magnetic charge. For this time-independent solution the electric field vanishes. At infinity the magnetic field is radial:
since it is obtained from the vacuum configuration <</ Its flux through the surface is
v through a gauge transformation.
1 1 4n B dS=- - r 2 dQ=-=g e r2 e
(12-213)
by definition of the magnetic charge g contained inside the sphere. Solutions of higher topological charge n carry a magnetic charge 4nnje. As any semiclassical configuration of this kind, the energy of the configuration-i.e., the rest mass of the monopole-is proportional to the inverse square coupling constant, Ije 2 't Hooft has shown that it is of order M w ja where M w is the vector mass acquired through spontaneous symmetry breaking. Such a monopole would be extremely heavy!
NONABELIAN GAUGE FIELDS
12-5-4 Renormalization of Spontaneously Broken Gauge Theories
It is clear that the gauge (12-209), the so-called unitary gauge as it involves only
physical degrees of freedom, is not suited for the study of renormalization, since the propagator has a bad large-momentum behavior. We need to use the initial gauge where renormalizability is more obvious, but then we will have to show that unphysical states do not contribute to the S matrix. The key idea is the same as in the case of the spontaneously broken (J model (Chap. 11). Renormalization is independent of whether the symmetry is exact or spontaneously broken. For simplicity of notations, we present the analysis in the case of G = O(n) and for a real scalar vector multiplet l/J. The complete lagrangian including gauge and ghost terms reads
Sl.f<
(,,2)
= _ "4 FI'va p"V a _ 2 (0 I' AI'a )2 1 ~
'/ I'
Dl'n
+ 'Z D1''1' DI'A-. _ p 'I' _ 1 A-. A-.2 'I' 2
A</> (A-.2)2 4 'I'
(12-214)
The cp4 coupling constant has been denoted A</> to avoid confusion with the gauge parameter. For J12 < 0, the field cp acquires a vacuum expectation value <cp> = v. We want to show that the counterterms of the symmetric theory (ji2 > 0) suffice to make finite the broken theory (J12 < 0) up to a renormalization of J1 2. As in Chap. 11, we cannot blindly continue J12 from positive to negative values, because J12 = 0 is not an analyticity point. A phase transition occurs at this point. It is safe, however, to introduce a small external source c coupled to the field l/J(x) and constant throughout space
(12-215)
This explicit breaking induces a vacuum expectation value v of l/J parallel to c. Indeed, an identity derived for the linear (J model [Eq. (11-170)J remains true here. If rt2 )(0) = - m 2 denotes the inverse propagator of the transverse component CPT at zero momentum, then c = -vq~)(O) (12-216) To lowest order, this condition reads c = (J12 + A</>V 2)V and expresses that the vacuum expectation value v = <l/J> makes the lagrangian (12-214) stationary. The proof of renormalizability relies on two observations, First, as in the linear (J model, the generating functional of proper functions for the symmetric and the broken theories are related by r(l/J, c, v) = rs(l/J with
+ v) -
rs(v)
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