w = 2L + 2 + L ni(d i - 2) :s; 2L + 2 in Visual Studio .NET

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w = 2L + 2 + L ni(d i - 2) :s; 2L + 2
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where ni is the number of vertices of type (i) and d i is the number of field derivatives on such a vertex. For a proper diagram with E external lines (among which no ~ line), this yields
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w :s; 2L + 2 - E
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(12-192)
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in the original theory (12-187). Observe that for an abelian theory, p.
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= il~~,
the fields
are not coupled to the vector
612 QUANTUM FIELD THEORY
-{-!"\~
""-'-~--"/
Figure 12-11 A diagram with a quartic divergence, in a massive gauge theory. The dotted lines represent auxiliary field I; propagators.
__ -e-__
field. We have just recovered the result of Chap. 8, namely, that massive electrodynamics is renormalizable by power counting. In a nonabelian case, if we restrict ourselves to the one-loop approximation, Eq. (12-192) gives the same superficial degree of divergence as in a renormalizable theory, OJ = 4 - E. To this order an effective lagrangian for diagrams without external I; or 11 lines reads in the Landau gauge
fE'l
= tr (iF"vPV -
M2 A"A" - : : 0"1; D"I; ) -
if0"D"11
(12-193)
The gaussian integrals over I; and 11, if may be performed. The former yields deC 1/2 .H j ' while the latter gives det .Hj'. Hence, to that order, a single auxiliary field suffices, with the prescription that a factor -i is attached to each closed ghost loop. The presence of this factor -i to be compared with a factor -1 in the massless case shows that the limit M ..... 0 must be singular. We have seen in Sec. 12-3 that the ghost contribution (with a factor -1) was crucial in the massless case to maintain gauge invariance. We thus expect the modification of the prescription to modify the counterterms. For instance, it leads to mass or gauge term renormalizations, and, more seriously, to other four-point couplings. Therefore, even though the theory looks renormalizable to this order, the symmetry of the counterterms is lost and serious difficulties occur to higher orders. For instance, according to Eq. (12-192) the diagram depicted on Fig. 12-11 has a quartic divergence, OJ = 4. We thus conclude that in spite of cancellations of divergences, the massive gauge theory is not renormalizable.
12-5-3 Spontaneous Symmetry Breaking
We have studied spontaneous symmetry breaking in Chap. 11, where boundary conditions allow us to choose among a set of degenerate ground states. A remarkable feature of this phenomenon, in the case of a continuous symmetry, is the appearance of massless particles. These Goldstone bosons are the zero-energy excitations connecting the possible vacua to each other. It is natural to reexamine this phenomenon in a gauge theory (abelian or nonabelian) where long-range forces are present or, alternatively, where there may exist an unphysical sector in the Hilbert space. It turns out that in the presence of a broken gauge symmetry, the long-range forces are screened. The Goldstone bosons and the gauge fields conspire to create massive excitations, and the massless excitations are unobservable. This phenomenon was discovered and studied in the context of superconductivity. Electron pairs responsible for superconductivity may be described by a wave function t/J = peie/~. The charge density, proportional to t/J*t/J = p2, must be constant throughout the crystal to neutralize the background charge of the ions. In the presence of a magnetic field with vector potential A, the current J reads J = -1 t/J* (I/z~ - 2qA -:- V 2m I
) t/J = -p2 (VB m
NONABELIAN GAUGE FIELDS
where q = 2e is the charge of the pair. The divergence of J vanishes, and hence in the transverse gauge V' A = 0 we have f).(} = O. For a simple geometrical configuration, () is constant; thus the Maxwell equation
f).A = -qJ
leads to (12-194) The vector potential A is screened on a characteristic length A, A = qp [. This is the Meissner effect which prevents a magnetic field from penetrating a superconductor. Returning to field theory consider a charged field c/J coupled to an abelian gauge field. The lagrangian is
fo /[
= -i-(oIlA v - ovAIl)(OIlA V- OVAIl) + (Oil - ieAIl)c/J*(OIl + ieAIl)c/J - V(c/J)
(12-195)
The potential V(c/J) is invariant under local transformations c/J -'> eiW(X)c/J and its minimum occurs for a nonzero value of c/J*c/J (compare with Fig. 11-5). For instance,
V(c/J)
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