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where p = 2 [q (x i for nn due to the identity of particles) is the phase space factor. This amplitude exhibits a pole at s "" M2 - ipaq2g2/16n corresponding to a resonance of width 1 "" paq2 g2 /16nM [s ~ M'. Adjusting the value g2/16n = 0.63 yields Ip "" 130 Me V, 1 K' "" 38 MeV, and I", "" 4.5 MeV, which reproduces fairly well the experimental values (125,50,3.2 respectively).
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12-5-2 Massive Gauge Theory
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Is a gauge theory where mass terms are introduced by hand renormalizable In electrodynamics, the situation is favorable. After separation of the gauge field into transverse and longitudinal components, the longitudinal part kl'kv/M2 which gives rise to the bad behavior in the propagator does not contribute to the S matrix. This results from the non interaction of longitudinal and transverse components and from the coupling of the field to a conserved current. In a nonabelian theory, none of these properties is satisfied. Longitudinal and transverse parts do interact, while the current to which the gauge field is coupled is not conserved. On the other hand, unexpected cancellations of divergences at the one-loop level make the theory look like renormalizable. This explains why it took some time to reach a consensus, namely, that the theory is not renormalizable. The way out of this unpleasant situation is to appeal to the mechanism of spontaneous symmetry breaking, to be explained in the next subsection.
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The aim is the construction of a renormalizable theory with the requirement that the physical states be the massive vector fields only. If we use auxiliary fields, as in the Stueckelberg method for the electromagnetic field, we must check that only the three physical degrees of freedom of each massive vector boson contribute to unitarity. In the forthcoming method, where local changes of variables will be performed in the functional integral so as to improve the behavior of the propagator, this requirement will be fulfilled by virtue of the equivalence theorem. We thus consider the generating functional
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[Sf - 2 tr (J. A)]}
(12-186)
where we use again matrix notations (12-187) The canonical quantization may be in trouble, since nO a = C5Sf/C5o oAo a == O. However, the presence of the mass term insures the existence of a propagator, and hence the definiteness of the previous functional integral in a perturbative sense. The Faddeev-Popov operation is not necessary, but we do it nevertheless, in order to improve the behavior of the propagator. We choose a gauge condition
ff(A) = C
NONABELIAN GAUGE FIELDS 611
as in Sec. 12-2-2 and insert in (12-186) the identity
We obtain
eG(J) =
f~(g) I]
O[ff(9A) - C] det A.,,(A)
(12-188)
f~(A)~(g) I]
o[ ff(9 A) - C] det A so exp {i
d4 x [2 - 2 tr (J. A)]}
In contrast with the massless case, 2 is no longer invariant under the gauge transformation A ..... 9 A. If we parametrize g(x) as
g(x) =
e<':(x)
it is easy to show that
where
stands for the formal series
P~(A, ~) =
+ 2)!
[... [D ~, ~], ], ... , ~]
(12-189)
the generic term of which has n brackets. The S matrix is not affected if we replace the source term J . A by J. q A. After a change of variable A ..... g-1 A and a gaussian integration over C, the new generating functional reads
eG'(J)
f~(A, ~,
if) exp {i
d4 x [2 '(A,
~, 1], ij) -
2 tr (J. A)] }
(12-190)
It involves a lagrangian with the field A., the Faddeev-Popov ghosts conventional commutation assignments
ij, and a new field
with
2 '(A,~, 1], ij) =
[Y.,P' -
A.A~ + -1.ff2(A) + 2 ~2 A il.~ - ~: il.~P (A, ~)J
ijA3"1]
(12-191)
If M2 = 0, the field ~ disappears from the lagrangian and the ~ integration gives an (infinite) factor which does not contribute to eG'(J) - G'(O). It is convenient to choose the Landau gauge, ff = il.A~, -1. ..... 00, to compute the vector propagator -i(g~,_k.k,lk2)(k2_m2)-I. The ~ and 1] propagators behave as liP. The superficial degree of divergence of an L-loop diagram is
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