.net pdf 417 reader Moreover, the Faddeev-Popov ghost also acquires a mass. The operator Aab reads in VS .NET

Print PDF417 in VS .NET Moreover, the Faddeev-Popov ghost also acquires a mass. The operator Aab reads

Moreover, the Faddeev-Popov ghost also acquires a mass. The operator Aab reads
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Aab(x, y)
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{OIlDllab - g; (v, T a Tb(cp'
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(12-224)
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since it results from an infinitesimal gauge transformation in g; acting on both A and cp'. The ghost-mass matrix is therefore
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(m 2)ab = gA (Tav, TbV)
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(12-225)
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Finally, this choice diagonalizes the quadratic form in A and cp'. The crossed term in the expansion of -(A/2)g;2 just cancels -g(oIlCP', AllaTav) arising from i(DIlCP, Dllcp). It follows that in terms of the mass matrix of Eq. (12-210), the vector propagator reads
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~IlV(k) =
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k2 _
~; + iB [gllv -
(1 - A- 1) k 2 _ A
~:~2 + iBJ
(12-226)
QUANTUM FIELD TIIEORY
As A ~ 00, one recovers the Feynman rules in the transverse (Landau) gauge, while as A ~ 0, all the unphysical masses recede to infinity. In the latter case, we do not expect these states with enormous masses to contribute to the S matrix. We proved in Sec. 12-4-4 that the S matrix does not depend on the choice of gauge. The argument which was formal due to the infrared divergences is now justified. We conclude that in any gauge, and in particular in the Landau gauge, the unphysical states do not contribute. A careful analysis should pay proper attention to renormalization. On this point the reader is referred to the literature.
Even though unphysical particles have disappeared from the physical subspace, there remains some trace of the spontaneous breaking mechanism, namely, the (massive) components of the scalar fields. Besides these scalar Higgs fields, we recall that some components of the vector field may remain massless. We may wonder whether it is mandatory to introduce scalar fields and whether it is not possible to generate them as bound states, for instance, of a fermion-antifermion pair. Such a dynamical breakdown is illustrated by the Schwinger two-dimensional massless electrodynamics. The vacuum polarization has a pole at zero momentum, the fermions disappear from the theory, and the only remaining single particle state is a bosonic bound state of mass e/Jn. In spite of the appeal of such a mechanism, it is not presently known how to realize it in four dimensions.
12-6 THE WEINBERG-SALAM MODEL
We present a realistic unified model of weak and electromagnetic interactions proposed independently by Weinberg and Salam and based on a spontaneously broken gauge theory. Among all the models of this type, it may be singled out because of its anteriority, its economical number of parameters, and the fact that it has received some experimental confirmation with the discovery of neutral currents and of charmed particles.
12-6-1 The Model for Leptons
The electron and its neutrino Ve are treated on the same footing as the muon and its neutrino vI'" The left helicity component of the charged lepton eL = (1 - ys)ej2 [IlL = (1 - ys),uj2] and its neutrino ve(v) are grouped into a column matrix (12-227) This suggests the introduction of a group of leptonic isospin for which Le and LI" are doublets, while the right components eR = (1 + ys)ej2 == Re and ,uR == RI" are singlets. A leptonic hypercharge Y is also assigned to each of these fields in such a way that the analog of the Gell-Mann and Nishijima rule is satisfied:
The left doublets have Y
T3+~
(12-228) 2. The weak isospin
-1 and the right singlets Y
NONABELIAN GAUGE FIELDS 621
T and hypercharge Y commute; therefore the transformation group is
SU(2) x U(l).
We then construct a gauge theory with this in variance group, involving a triplet of gauge fields AI' for SU(2) with a charge g and a field BI' for U(l). The U(l) coupling constant will be denoted g'/2. Since we want a single gauge field (the photon) to remain massless after spontaneous breaking, we introduce a doublet of complex scalar fields:
(12-229)
of hypercharge Y is
+ 1. The most general renormalizable invariant
potential for
(12-230)
For f12 < 0, acquires a nonvanishing vacuum expectation value, which may be assumed real, along o:
=_1 (0)
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