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The axial current is defined as $$j^\mu_5 = \bar{\psi} \gamma^\mu \gamma_5 \psi.$$ This quantity is important when studying anomalies. Explicitly working out components, the axial current is just the current due to left-handed fermions minus the current due to right-handed ones, just like the usual current is their sum.

The above is true just by definition, but it leaves me unsure what the axial current and the associated charge are.

  • Is there a nice, physical interpretation of $j^\mu_5$ besides the one I gave above?
  • How does one measure the axial current/charge experimentally?
  • What is the physical meaning of a time-ordered correlation function of $j^\mu_5$'s?
knzhou
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1 Answers1

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  1. The axial (also called "chiral") current is the Noether current for the axial transformation $$ \psi(x) \mapsto \mathrm{e}^{\mathrm{i}\alpha \gamma^5}\psi(x),\alpha\in\mathbb{R}$$ of a massless fermionic theory with the usual Dirac action $$ S_D[\psi] = \int \bar\psi(x)\mathrm{i}D_\mu\gamma^\mu\psi(x)\mathrm{d}^d x$$ so it is classically non-conserved only by the transformation of the mass term in a massive Dirac action.

  2. One experimental significance of the non-conservation of the axial current is neutral pion decay, see this question and references therein. Heuristically, it's the axial current and not the usual current that plays a role there because the pion is a pseudoscalar, not a scalar, and hence must couple to another pseudo-object (which the axial current is) to give a scalar amplitude. By explicit calculation, you find that the chiral anomaly is directly related to the decay width of the pion.

  3. No idea. :)
Community
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ACuriousMind
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