Chiral symmetry

From the above, we therefore conclude that the nanotube modes obtained by setting n = N/2, transform according to the B irreducible representation of the chiral symmetry group G. 

Keywords QCD, instanton, vacuum, quark, action, chiral symmetry. 

The presence of instantons in QCD vacuum very strongly affects light quark properties, owing consequent generation of quark-quark interactions. These effects lead to the formation of the massive constituent interacting quarks. This implies spontaneous breaking of chiral symmetry (SBCS), which leads to the collective massless excitations of... 

Note, that we studied (Nasriddinov, 1998) the t —> ir riuT decay of the r lepton in the framework of this method with taking into account the isotopic spin violation of chiral symmetry in the Oakes scheme.In this case the 7r° — rj - mixing Lagrangian has the form... 

The second issue is how to explain the observation of both left- and right-handed helices in the phosphonate material. While Thomas et al. found both helical senses in the early stages of formation of DCggPC tubules, they found both helical senses even in the equilibrium state of the phosphonate. In the previous section, we attributed their results on tubule formation kinetics to a biased chiral symmetry-breaking in which the molecular packing has two possible states which are approximately mirror images of each other. The... 

So far we have considered the formation of tubules in systems of fixed molecular chirality. It is also possible that tubules might form out of membranes that undergo a chiral symmetry-breaking transition, in which they spontaneously break reflection symmetry and select a handedness, even if they are composed of achiral molecules. This symmetry breaking has been seen in bent-core liquid crystals which spontaneously form a liquid conglomerate composed of macroscopic chiral domains of either handedness.194 This topic is extensively discussed in Walba s chapter elsewhere in this volume. Some indications of this effect have also been seen in experiments on self-assembled aggregates.195,196... 

Following the notation of Eq. (9), a general free energy for chiral symmetry -breaking in membranes can be written as... 

Here, the final three terms are a Ginzburg-Landau expansion in powers of i j. The coefficient t varies as a function of temperature and other control variables. When it decreases below a critical threshold, the system undergoes a chiral symmetry-breaking transition at which i becomes nonzero. The membrane then generates effective chiral coefficients kHp = k n>i f and kLS = which favor membrane curvature and tilt modulations, respec-... 

There are two reasons to think this situation might occur. The first reason is experimental. As discussed in Sections 2-5, in most experiments on chiral materials, tubules and helical ribbons are observed with only one sense of handedness. However, there are a few exceptions in experiments on diacetylenic phospholipids,144 diacetylenic phosphonate lipids,145 146 and bile.162 In these exceptional cases, some helices are observed with the opposite sense of handedness from the majority. In the work on diacetylenic phospholipids, the minority handedness was observed only during the kinetic process of tubule formation at high lipid concentration,144 which is a condition that should promote metastable states. Hence, these experiments may indeed show a case of biased chiral symmetry-breaking in which the molecular chirality favors a state of one handedness and disfavors a mirror image state. 

We close our theoretical discussion with the concept of biased chiral symmetry-breaking because it sums up this speculation about the interplay... 

Figure 8.20 Structure and phase sequence of prototypical bent-core mesogen NOBOW (8) are given, along with space-filling model showing one of many conformational minima obtained using MOPAC with AMI force field. With observation by Tokyo Tech group of polar EO switching for B2 smectic phases formed by mesogens of this type, banana LC field was bom. Achiral, polar C2v layer structure, with formation of macroscopic spontaneous helix in polarization field (and concomitant chiral symmetry breaking), was proposed to account for observed EO behavior.

There is also a topological term which is essential in order to satisfy the t Hooft anomaly conditions [32-34] at the effective Lagrangian level. It is important to note that respecting the t Hooft anomaly conditions is more than an academic exercise. In fact, it requires that the form of the Wess-Zumino term is the same in vacuum and at non-zero chemical potential. Its real importance lies in the fact that it forbids a number of otherwise allowed phases which cannot be ruled out given our rudimentary treatment of the non-perturbative physics. As an example, consider a phase with massless protons and neutrons in three-color QCD with three flavors. In this case chiral symmetry does not break. This is a reasonable realization of QCD for any chemical potential. However, it does not satisfy the t Hooft anomaly conditions and hence cannot be considered. Were it not for the t Hooft anomaly conditions, such a phase could compete with the CFL phase. 

The validity of the t Hooft anomaly conditions at high matter density have been investigated in [32, 33], A delicate part of the proof presented in [33] is linked necessarily to the infrared behavior of the anomalous three point function. In particular one has to show the emergence of a singularity (i.e. a pole structure). This pole is then interpreted as due to a Goldstone boson when chiral symmetry is spontaneously broken. 

A. Mocsy, F. Sannino and K. Tuominen, Confinement versus chiral symmetry, Phys.Rev.Lett.92 182302 (2003) [arXiv hep-ph/0308135],... 

Relativistic mean-field model based on 517(3) chiral symmetry [53] ( XSU(3) ), see Fig. 5... 

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