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cos 8 e Figure 12-15 KO
---> fl.+ fl.-
decay.
NON ABELIAN GAUGE FIELDS
As above, T a is some component of the weak isospin and Y denotes the weak hypercharge. This corresponds, for instance, to triangle A a A aB diagrams. It folio ws that leptons give a nonvanishing contribution to the anomaly. With the two doublets Le and LI' of (12-227), we have
leptonic doublets
Y = 2 x (-1)
(12-264)
However, if we incorporate hadrons according to the previous scheme, the two doublets of (12-261) contribute
hadronic doublets
Y=2xj
(12-265)
If, moreover, we assume-as in the discussion of the n decay in Chap. II-that quarks come in three degenerate unobservable colors, the expression in (12-265) must be supplemented by a summation over color, i.e., multiplied by a factor 3. Leptonic and hadronic contributions to the anomaly thus cancel. We emphasize the importance of the equal number of leptonic and hadronic doublets-for the previous weak isospin and charge assignments-and of the color degeneracy factor for the validity of this argument. The computation of the n decay is not affected by the present cancellation mechanism. Indeed, recall that n is coupled to the divergence of the third component of the ordinary axial isocurrent, that is, (ii, d)YI'Y5'[3/2(~). We have omitted so far the Yukawa-like couplings of the scalar mesons to quarks and the quark-mass terms. Before spontaneous breakdown takes place, the theory must be SU(2) x U(l) invariant, which forbids quark-mass terms. However, after spontaneous breaking and shift of the Higgs field, the quarks acquire a mass
fern =
_1i/
~ Y5 AI/I_1i/1 ~ Y5
Atl/l
(12-266)
Here A is a matrix which is a priori neither diagonal nor real, and is just restricted to commute with the charge operator Q. By independent unitary redefinitions of the left- and right-handed components, it is possible to diagonalize it:
1/1' L= U LI/IL
I/I'R
U RAul
URI/IR
(12-267)
M diagonal (12-268)
The eigenstates 1/1' of the mass matrix are the quarks u, d, s, c. On the other hand, the charged current which reads (12-269)
QUANTUM FIELD THEORY
becomes (12-270) where U1L and U2L are the restrictions of UL to the upper (u, c) and lower (d, s) components respectively. By a further redefinition of the relative phases between the quarks [of course, unobservable on (12-268)J this may be cast into the form
(uc)y (1 _
5) ( co.s Be Y _ sm Be
sin Be)(d) cos Be S
(12-271)
in agreement with (12-263). We conclude that the Cabibbo angle comes from the mismatch between the eigenstates of the mass matrix and the quark components entering the charged current.
A presentation of models of weak interactions should include a review of their implications-on neutrino scattering off hadrons, in particular. It is wiser to refer the reader to more competent authors for a thorough discussion. In view of the recent experimental discoveries, the preceding theoretical framework may and must be extended to include more quarks and more leptons. We content ourselves with a simple remark. The Cabibbo mixing matrix which took the simple form (12-271) in the case of four quarks may depend on more parameters, some of which may be complex, thus introducing CP violations. The recent years have seen a blossoming of theoretical models, involving various groups, multiplet assignments, possible right-handed couplings, etc. Any discussion is doomed to become obsolete very soon. How would strong interactions enter this scheme For reasons to be discussed in the next chapter, a nonabelian gauge theory of strong interactions nowadays seems to be a good candidate. The gauge group would be a SU(3)c group, unrelated to the GeIl-Mann and Ne'eman octet symmetry. The eight-gauge fields-the so-caIled gluons-would be coupled to the color quantum numbers of the quark triplets. In contrast with the spontaneously broken symmetry of weak and electromagnetic interactions, this local SU(3)c symmetry would be exactly implemented and the g1uon would remain massless. In this gauge description of weak, electromagnetic, and strong interactions, quarks carry the two quantum numbers of color and flavor. On the other hand, vector bosons W, Z, ... , or the photon are colorless, while the gluons have neither flavor nor charge. FinaIly, it is possible to speculate that the group Gw x SU(3)c [G w = SU(2) x U(1) for the Weinberg-Salam model] originates from the breaking of a larger simple group. Such a superunification might even extend to gravitation.
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