20191201_004133394.pdf - ¯ µγ5ψ j5µ = ψγ µ ¯ 5ψ...

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U (1) A j μ 5 = ¯ ψγ μ γ 5 ψ μ j μ 5 = 2 im ¯ ψγ 5 ψ Q 5 = ψ γ 5 ψ = N L N R . (1.8) Therefore, the massless free theory is U (1) A invariant. As j μ and j μ 5 are both conserved, the following currents and charges j μ L = 1 2 ( j μ j μ 5 ) j μ R = 1 2 ( j μ + j μ 5 ) Q L = N L Q R = N R (1.9) are preserved, and the number of left-handed and right-handed fermions is constant over time. What happens if we switch on a gauge field? As our transformation deals only with the fermionic degrees of freedom, the naive conclusion would be that the conservation of both currents holds. Nevertheless we know that there are gauge contributions to the anomaly, which can be computed in perturbation theory [2]. In fact, the so-called triangle diagrams Figure 1.2: Example of triangle diagram contributing to the anomaly. give rise to the equation μ j μ 5 = 2 im ¯ ψγ 5 ψ N F g 2 16 π 2 ǫ αβμν F c αβ F c μν . (1.10) The new term P = N F g 2 16 π 2 ǫ αβμν F c αβ ( x ) F c μν ( x ) (1.11) is the Pontryagin density P , which encodes the topological properties of the Yang-Mills potentials and fields , i.e., defects, dislocations and instanton solutions of the gauge fields, deeply related to the winding number of the configurations. Thence, chiral symmetry is only preserved at the classical level, for the quantum corrections of the fermion triangle diagrams break it explicitly. This phenomenon is known as the Adler-Bell-Jackiw anomaly , honoring the discoverers [2]. The path integral formulation Z = integraldisplay bracketleftbig dA a μ bracketrightbig dψd ¯ ψ e R d 4 x L (1.12) can also give account of the anomaly. Although the massless action is invariant under a chiral transformation, the integral measure is not, and its Jacobian J = e i R d 4 ( x ) N F g 2 16 π 2 ǫ αβμν F c αβ ( x ) F c μν (1.13) reproduces exactly the anomalous contribution of the Adler-Bell-Jackiw triangle diagrams. 6
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xv Figure 5.9: Numerical performance for the (20,20,8) DGS mesh for a Poiseuille flow 129 Figure 5.10: Schematic of the Couette flow problem 130 Figure 5.11: Surface mesh for the Couette flow and internal points 131 Figure 5.12: DGS meshes for Couette flow: (15,15,5) and (20,20,8) 133 Figure 5.13:
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