NONABELIAN GAUGE FIELDS in VS .NET

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NONABELIAN GAUGE FIELDS
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12-6-4 Incorporation of Hadrons
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A natural way to incorporate hadrons in the previous scheme is to couple quark multiplets to the gauge fields. The structure of the charged hadronic current in terms of the conventional quarks u, d, S has already been given (Sec. 11-3). It contains strangeness conserving and strangeness changing components, with a Cabibbo mixing angle. This angle could perhaps be predicted in the framework of a gauge theory of weak interactions. In addition to this charged current, a neutral current coupled to the Z boson emerges in theories like the WeinbergSalam model. The constraint that strangeness changing processes induced by this neutral current should not be in violent contradiction with experiment turns out to be a stringent demand, and requires the introduction of at least a new quark. We start with the three quarks u, d, S and neglect for the time being their couplings to the Higgs scalars. The Cabibbo angle 8c is assumed to be given. The usual charged hadronic current will be reproduced if we give the following assignments to the quark components:
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N L == (d8 == d cos
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(s cos
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(12-257)
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is a doublet of weak isospin, with Y = j, while UR, dR, SR, and S8L == 8c - d sin 8c )L are isosinglets, with Y =~, -~, -~, -~. The interaction
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(g~.~ + ~ ;' $) NL + [S8L( - ~ ~ )S8L + (S8L-+d R) + (S8L-+SR)]
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(12-258)
+ ~URg'~UR + hc) + eA/lj~m -
may be rewritten in terms of the fields W, A, and Z of Eqs. (12-236) and (12-238) as
2 int
2fl(W ,th/l
J g2 + g,2 Z/l(NL ~ Y/lNL -
sin 2
8wj~m)
(12-259)
with
h/l = uY/l(l - Ys)(d cos 8c + S sin 8c)
= ~uY/lU - j(dY/ld + sY/ls)
as expected. The neutral boson Z is coupled to the electromagnetic current and to
N L'f3Y/lN L = ULY/lUL - cos 2 8c d LY/ldL - sin 8c cos 8c (d LY/lSL
+ hY/ld L)
- sin 2 8c s LY/lSL The terms proportional to sin 8c cos 8c are the strangeness changing neutral currents. They are embarrassing because experimentally I1S = 1 or I1S = 2, I1Q = 0 processes are heavily suppressed. For instance,
T'(K : n vv) = 6 x 10-7 T'(K - -+ all) T'(KO -+ J1+ J1-) :::::: 10-8 T'(KO -+ all)
(12-260)
QUANTUM FIELD THEORY
This puzzle was solved by Glashow, Iliopoulos, and Maiani (1970) through the introduction of a fourth quark, denoted c, of charge j like u and carrying a new quantum number, the charm. The left-handed component CL is supposed to form an isodoublet, together with the combination SOL = -d L sin Be + SL cos Be. In other words, there are now two left-handed doublets with Y = jand four right-handed singlets UR, dR, SR, and CR with Y = respectively. It is easy to see that the neutral boson is now coupled to
(ii, doh
(12-261)
-j, -j, ~
~ YI'(~)L + (c,soh ~YI'C:)L -
Bwj~m
(12-262)
Strangeness changing neutral currents occur in the first and in the second terms of (12-262) but cancel exactly. This means that the unwanted currents have been eliminated to order G. However, strangeness changing neutral transitions induced by higher-order exchanges of charged currents might still endanger the theory. This is the case of the diagrams drawn in Fig. 12-15, expected to be of order Grx and hence in disagreement with the very low experimental rates (12-260). However, after introduction of the C quark, the form of the charged current becomes
(12-263)
When all contributions are taken into account, it turns out that the dangerous amplitude is proportional to (m; - m~)/Mfir. For charmed quarks much lighter than the W meson, that is, me S; 1.5 - 2 GeV, there remains no discrepancy with experiment. Besides this property of suppressing the I1S =1= 0 neutral currents, the charmed quark has also an aesthetical appeal. It restores a symmetry between the four leptons and the four quarks, and, as a bonus, it removes the anomalies of the Weinberg-Salam model. Indeed, the lepton model studied in the previous sections is plagued with chiral anomalies. If we apply the analysis of Sec. 12-4 with only weak isosinglets and doublets, we readily discover that the only non vanishing anomaly is proportional to y tr (raTay) IX.
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