Triangle anomalies

We should mention that strictly speaking the theory as presented is not renormalizable, despite our pretending it was so, because of a technical complication known as a triangle anomaly. It will tmn out, surprisingly. 

We further find that it is necessary to endow eeich quark with a completely new, internal degree of freedom ceiUed colour . Eeich quark can exist in three different colour states. Severed technical problems— the statistics of quarks, the tt decay rate, and the existence of triangle anomalies— are thereby resolved. The implications of colour for the famous ratio... 

This unwelcome discovery is potentially catastrophic for our unified weak and electromagnetic gauge theory. There we have lots of gauge invariance, many conserved currents, both vector and axial-vector, and hence many Ward identities. Moreover the Ward identities play a vital role in proving that the theory is renormalizable. It is the subtle interrelation of matrix elements that allows certain infinities to cancel out and render the theory finite. Thus we cannot tolerate a breakdown of the Ward identities, and we have to ensure that in our theory these triangle anomalies do not appear. 

Although the data (mostly from neutrino experiments) on the c—s coupling are not conclusive, the evidence suggests that it is also left-handed. The apparent connection between quarks and leptons is fascinating. Both sets of doublets are point-like s = 5 fermions their electric charges are quantized in a related way (Q, Q — 1 with Q = 0 for leptons and Q = for quarks) furthermore, the sum of the electric charges of all fermions (in a colour scheme) vanishes as required to cancel triangle anomalies (see Section 9.5.3). It should be noted, however, that serious questions arise if one tries to put the above quark-lepton connection on a quantitative... 

Figure 5 Relationship among loci of structural, dynamic, and thermodynamic anomalies in SPC/E water. The structurally anomalous region is bounded by the loci of q maxima (upward-pointing triangles) and t minima (downward-pointing triangles). Inside of this region, water becomes more disordered when compressed. The loci of diffusivity minima (circles) and maxima (diamonds) define the region of dynamic anomalies, where self-diffusivity increases with density. Inside of the thermodynamically anomalous region (squares), the density increases when water is heated at constant pressure. Reprinted with permission from Ref. 29.

Figure 3.5 Radically anomalous noble gas isotopic compositions in extrasolar materials isolated from undifferentiated meteorites (from Anders Zinner, 1993). Stepwise heating of whole-rock meteorites liberates slightly more or less of components such as Xe-HL and Ne-E, relative to other noble gas reservoirs in the rock, leading to the modest isotopic variations (e.g., Xe compositions as illustrtated in Figure 3.4, or Ne compositions to the lower-left of the air-spallation-solar wind triangle in Figure 3.3) from which the presence of anomalies was originally inferred.

The most noteworthy feature of the sulfenylation and selenenylation rates (represented by the triangles) is their much diminished sensitivity to substitution. This reflects both smaller electron demand in the TS and increased sensitivity to steric factors. The relatively low rate of styrene toward selenenylation is somewhat of an anomaly, and may reflect both ground state stabilization and steric factors in the TS. The epoxidation data (CH3CO3H, hexagons) show a trend similar to bromination, but with a reduced slope. There is no evidence of a rate-retarding steric component. One indicator of a strong steric component is decreased reactivity of the E-isomer in an E,Z—disubstituted alkene pair, but the rates for the 2-butene isomers toward epoxidation are very similar (Table 5.9). 

Figure 3.9 Plots of the Stokes shift versus open triangles assign the anomalies, which Hammetta-constantsdifferenceofthetwo4,4 - persist only in polar solvents, and occasionally positioned stilbene substituents (X and Y) may form a third group of stilbenes. The (taking into account their relative sign) in uncertainty in estimation of A /2.3kTwas found...

Fig. 49. Upper critical field 3 2 and magnetic anomaly line 5 obtained from specific heat (squares) and thermal expansion (triangles) experiments (Hellfich, 1996 Kromer et al., 1998) for a high Tc sample.

Figures. Phase diagram ofthe Yoshida-Kamakura potential for a = 3.3. PandT areinreduced units. Full dots are two-phase coexistence points. Open dots are points of density maximum in the fluid phase. Diamonds and triangles denote points of —S2 maxima and D minima, respectively (D being the self-diffusion coefficient), giving the left boundary of the regions of structural and diffusion anomaly (the right boundaries, which are defined by —52 minima and D maxima, are out of the T range shown). Data are from Ref. [88].

Fig. 21. Phase diagram for superconducting UPtj with external field H along c. Open and solid circles are ftorn a.c. susceptibility data, while triangles are locations of torsional oscillator anomalies (Kleiman et al. 1989). The shaded area represents the boundary between states where antiferromagnetic intensity varies with field and temperature and where it is H and T independent. The dashed lines are trajectories followed to accumulate dilTraction data. (From Aeppli et al. 1989.)...

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