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This chapter opens the discussion of physical phenomena in the vacuum with Fermi points. The non-trivial topology in the momentum space leads to anomalies produced by the massless chiral fermions in the presence of collective fields such as magnetic field, textures, and vortices. In ³He-A, this gives rise to the anomalous mass current, the paradox of the orbital angular momentum, etc. All these phenomena are of the same origin as the chiral anomaly in relativistic quantum field theories. For example, the momentum exchange between superfluid vacuum and quasiparticles in ³He-A is analogous to electroweak baryoproduction — formation of excess of matter over anti-matter due to chiral anomaly. This allowed for the measurement of the spectral-flow force acting on a vortex-skyrmion in superfluid in ³He-A to experimentally verify the Adler–Bell–Jackiw equation describing chiral anomaly in particle physics. The chapter also discusses the Novikov–Wess–Zumino action, the interplay between r-space and p-space topology relevant for the discussed phenomena, and introduces 3 reactive forces acting on a vortex in fermionic superfluids: Magnus, Iordanskii, and Kopnin forces.

There are four forces acting on a vortex moving with respect to normal and superfluid components of the liquid: three reactive forces (Magnus, Iordanskii, and Kopnin forces) and the Stokes friction force. The Iordanskii force comes from the analog of the gravitational Aharonov–Bohm effect and are discussed later in the book. The Kopnin or spectral flow force appears only in fermionic superfluids. This chapter discusses the Kopnin force acting on a singular vortex. The origin of this force is the phenomenon of chiral anomaly, but as distinct from the force acting on continuous vortex discussed in Chapter 18, the Adler–Bell–Jackiw equation for chiral anomaly is not applicable. The anomaly is now related to the spectral flow of fermion zero modes in the vortex core, which is caused by motion of the vortex with respect to the normal component. The spectral flow leads to generation of linear momentum by a vortex core, which is then transferred to the normal component. This process is similar to the generation of baryonic charge by the cores of cosmic strings and serves as the analog of Callan–Harvey mechanism of cancellation of anomalies. The temperature dependence of the spectral flow force is obtained from solution of kinetic equation for fermion zero modes. The resulting force acting on a vortex has been measured in experiments on vortex dynamics in ³He-B. Existence of two forces in addition to conventional Magnus and friction forces — Iordanskii and Kopnin forces – has been verified.

This chapter shows that a baryon asymmetry can be generated only if the Sakharov conditions are satisfied. The required departure from thermal equilibrium can be realised either in first-order phase transitions or in out-of-equilibrium decays. Sphaleron transitions as a possible way to transfer a lepton into a baryon asymmetry are covered and the Boltzmann equation governing the baryon asymmetry generated in particle decays is derived. A brief review of baryogenesis during the electroweak phase transition is presented.

Chapter 4 deals with the stability of the proton, hence of hydrogen, and how to reconcile that stability with the baryon number nonconservation (or baryon conservation) needed to establish a matter–antimatter imbalance in the infant universe. Sakharov’s three conditions for establishing a matter–antimatter imbalance are presented. Grand unified theories and experimental searches for proton decay are described. The concept of spontaneous symmetry breaking is introduced in describing the electroweak phase transition in the infant universe. That transition is treated as the potential site for introducing the imbalance between quarks and antiquarks, via either baryogenesis or leptogenesis models. The up–down quark mass difference is presented as essential for providing the stability of hydrogen and of the deuteron, which serves as a crucial stepping stone in stellar hydrogen-burning reactions that generate the energy and elements needed for life. Constraints on quark masses from lattice QCD calculations and violations of chiral symmetry are discussed.