Topological vector currents have gained interest recently with their possible verification at RHIC through the Charge Separation Effect and the Chiral Magnetic Effect. Much work has been done in understanding the role of topological vector currents in astrophysics, specifically in the interiors of neutron stars and quark stars. We will discuss a recent aspect of this work regarding pulsar kicks. A significant percentage of the pulsar population is known to have velocities above 1000 km/s, but a suitable explanation for these velocities does not exist. We will detail how topological currents may be responsible for these large kicks and discuss why the mechanism is successful where others fail.
A recent topic of much interest has been the P and CP-odd effects that arise from the axial anomaly. The most popular of these has been the Chiral Magnetic Effect [1], but this is part a body of work investigating this phenomenon that starts with topological currents in condensed matter systems [2], and includes the study of anomalous axion interactions in QCD [3], the Charge Separation Effect [4], and the high density analogue of the Chiral Magnetic Effect in dense stars [5,6]. The Chiral Magnetic effect is particularly exciting because it rests on the edge of observational science. The current may be responsible for the parity violating effects seen in the STAR collaboration at RHIC [7]. Here we will discuss how the existence of these currents in dense stars may be responsible for generating the large proper motion seen in some pulsars [8].
The goal of the paper [8] was to elaborate on a kick mechanism first discussed by [6] that may explain pulsar velocities greater than 1000 km s -1 . There have been a number of studies that have compiled and modelled the velocities of pulsars. Although they disagree on whether the distribution is indeed bimodal, they agree that a significant number of pulsars are travelling faster than can be attributed to neutrino kicks. The analysis of [9] favours a bimodal velocity distribution with peaks at 90 km s -1 and 500 km s -1 with 15% of pulsars travelling at speeds greater than 1000 km s -1 . Alternatively [10] and [11] both predict a single peaked distribution with an average velocity of ∼ 400 km s -1 , but point out that the faster pulsars B2011+38 and B2224+64 have speeds of ∼ 1600 km s -1 . Large velocities are unambiguously confirmed with the model independent measurement of pulsar B1508+55 moving at 1083 +103 -90 km s -1 [12]. Currently no mechanism exists that can reliably kick the star hard enough to reach these velocities. Asymmetric explosions can only reach 200 km s -1 [13], and asymmetric neutrino emission is plagued by the problem that at temperatures high enough to produce the kick the neutrinos are trapped inside the star [14]. Alterations of the neutrino model that take into account only a thin shell of neutrinos require large temperatures and huge surface magnetic fields.
We will provide a sketch of how the kick is generated and direct those interested in the details to read [8]. The kick mechanism we will discuss relies on the existence of topological vector currents of the form described by [6], which some readers may recognize as the same current responsible for the Chiral Magnetic Effect [1] in QCD,
where n R and n L are the one dimensional number densities of the right and left-handed electrons, and Φ is the magnetic flux. There are three requirements for topological vector currents to be present: an imbalance in left and right-handed particles µ L = µ R , degenerate matter µ T , and the presence of a background magnetic field B = 0. All of these are present in neutron and quark stars. The weak interaction, by which the star attains equilibrium, violates parity; particles created in this environment are primarily left-handed. The interior of the star is very dense, µ e ∼ 10 MeV, and cold, T ∼ 0.1 MeV, such that the degeneracy condition µ T is met, and neutron stars are known to have huge surface magnetic fields, B s ∼ 10 12 G.
If the electrons carried by the current can transfer their momentum into space-either by being ejected or by radiating photons-the current could push the star like a rocket. In typical neutron stars this is unlikely because the envelope (the region where µ ∼ T ) is thought to be about 100 m thick. Once the current reaches this thick crust, it will likely be reabsorbed into the bulk of the star. But if the crust is very thin, or nonexistent, the electrons may leave the system or emit photons that will carry their momentum to space. The electrosphere for bare quark stars is thought to be about 1000 fm. With this in mind we conjecture that stars with very large kicks, v 200 km s -1 , are quark stars and that slow moving stars, v ≤ 200 km s -1 , are kicked by some other means, such as asymmetric explosions or neutrino emission, and are typical neutron stars. Confirmation of this would provide an elegant way to discriminate between neutron stars and quark stars.
The total number current for electrons reaching the surface of the star is calculated in [8] and is given by,
where B c = 4.4 × 10 13 G is the critical magnetic field, T core is the core temperature of the star, and n 0 is nuclear density. The typical density for quark matter is n b ∼ 10 n 0 but could easily be higher. Though many pulsars have a surface field of around 10 12 G, the field in the bulk of the star is likely much stronger based on virial theorem arguments in [15],which yield possible core fields of B max ∼ 10 18 G. This is an extremely large field and is unlikely as it is a strict upper bound. Based on this we choose a value of the core magnetic field to be
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