as given by Eq. (11-225), is proportional to the combination in Visual Studio .NET

Generate PDF 417 in Visual Studio .NET as given by Eq. (11-225), is proportional to the combination

as given by Eq. (11-225), is proportional to the combination
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== tr (P{ T b, T
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(12-172)
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The vanishing of this quantity may be realized by each species of fermion (each representation) coupled to the gauge field; this is in particular what happens for real representations where the matrices Tare antisymmetric: (12-173)
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QUANTUM FIELD THEORY
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But the condition may also result from a cancellation between different species. This will be illustrated in Sec. 12-6-4. The harmful anomalies are those occurring in axial currents coupled to gauge fields. For instance, if the combinations Ii/Y/lYsAao/ are singlets for the internal group G, the anomalies of such currents are of no importance here. The matrices Aa might refer to a different set of quantum numbers, e.g., flavor as opposed to color. The corresponding anomalies are proportional to dabe = tr (A a{ T b T e}) = tr Aa tr {Tb, T e} , which vanishes in the case of an SU(3) symmetry, for instance, since tr Aa = Only the U(I) current
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(12-174) is anomalous:
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2imli/yso/
+ Cg 2c/lvp"F/lV FP"a a 28/lc/lvp"A V (8 P 2imli/yso/ + 2Cg a A"a
]/lS = J/l S
8" APa -1gCabe APbA"e)
(12-174a)
with C = - Tf /16n 2 On the other hand, there exists a conserved current
2Cg2c/lVp"Avi8PA"a - 8"APa -1gCabe APbA"e)
(12-175)
but it suffers from the lack of gauge invariance.
12-5 MASSIVE GAUGE FIELDS 12-5-1 Historical Background
The nonabelian gauge theories considered so far enjoyed an exact local symmetry, and consequently the gauge field was massless. Such theories are used nowadays to construct models of strong interactions. However, historically, after the introduction by Yang and Mills of nonabelian gauge fields, physicists have struggled for years in order to build a meaningful theory of massive gauge fields, hence breaking explicitly the local symmetry. A strong motivation came from the study of weak interactions. We recall from Chap. 11 that the current-current Fermi theory provides a remarkable phenomenological framework. The weak interaction lagrangian (or up to a sign, the hamiltonian) was written (compare with 11-62)
.Pint
= - j2J/l(x)J/lt(x)
(12-176)
This is, of course, a zero-range interaction. In spite of its successes for low-energy processes, this model suffers from serious problems. As the dimensionality of the coupling constant G shows, the theory is nonrenormalizable. Alternatively, power counting assigns dimension six
NONABELIAN GAUGE FIELDS
to the product JI"J/. At high enough energy we cannot content ourselves with the Born approximation. In order that a scattering amplitude satisfies the unitarity condition, at least perturbatively, higher-order terms must be added. These corrections, however, are plagued with ultraviolet divergences, the elimination of which introduces a growing number of arbitrary parameters. In practice, the nonrenormalizability makes this computation impossible. Another aspect of the same problem arises when we consider the Born approximation for some cross section (J. On dimensional grounds, we expect at high energy the behavior 2 (J ~ constant x G s (12-177) where s is the total center of mass energy square, while in every partial wave the unitarity limit reads constant (12-178) s
(J~---
Therefore, we expect a violation of unitarity to arise at energies of the order ~ G- 1/ 2 ~ 300 GeV.
It is a good exercise to compute explicitly the constant appearing in (12-177) and (12-178) for such
leptonic processes as vv ---> VV, Vee -
---> Vee - ,
and vpe -
---> vej.l- .
Both aspects, nonrenormalizability of the theory and bad high-energy behavior of the Born approximation, are manifestations of the same phenomenon. This is obvious if we use dispersion relations to compute a one-loop contribution to some elastic scattering amplitude in terms of its discontinuity, i.e., in terms of some Born cross section. The behavior of the latter results in severe divergences in the dispersion integral. It is therefore mandatory to transform the Fermi theory into a respectable, i.e., renormalizable, field theory. A tempting hypothesis consists in introducing a charged vector field Wil and coupling it to the current J Il :
.Pint
= gJll(x)Wllt(x) + hc
(12-179)
The analogy with electromagnetism is evident. The intermediate boson represented by the field W would be the quantum of weak interactions. To explain the validity of the Fermi theory at low energy, we assume W to be very massive. The implications of (12-179) would depart from those of (12-176) only at high energies. Let us consider, for instance, the f1 decay. The Fermi theory gives the amplitude (Fig. 12-9a)
u(pe)yp(l - ys)v(PvJu(Pv)yP(l - ys)u(Pll)
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