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

• No School
• AA 1
• 4

This preview shows page 1 - 3 out of 4 pages.

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
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:

#### You've reached the end of your free preview.

Want to read all 4 pages?

• Fall '19

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern